Special Functions

Special functionsFrom Wikipedia, the free encyclopedia

Contents

1 Abramowitz and Stegun 11.1 Editions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Successor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Absolute value 32.1 Terminology and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Absolute value function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Relationship to the sign function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.3 Antiderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5.1 Ordered rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.3 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Ackermann function 123.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Table of values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Use as benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

i

ii CONTENTS

3.7 Ackermann numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Airy function 194.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Asymptotic formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Complex arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4.1 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 Relation to other special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.6 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.7 Fabry–Pérot interferometer Airy Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Algebraic function 265.1 Algebraic functions in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.2 The role of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.3 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Anger function 316.1 Relation between Weber and Anger functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Arithmetic–geometric mean 337.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.5 Proof of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.6 Proof of the integral-form expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

CONTENTS iii

7.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Askey–Gasper inequality 378.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 List of special functions and eponyms 399.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.3 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.4 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.5 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.6 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.7 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.8 H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.10 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.11 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.12 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.13 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.14 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.15 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.16 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.17 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.18 W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.19 Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Special functions 4510.1 Tables of special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.1.1 Notations used for special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.1.2 Evaluation of special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.2 History of special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2.1 Classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2.2 Changing and fixed motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2.3 Twentieth century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.2.4 Contemporary theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10.3 Special functions in number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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10.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 1

Abramowitz and Stegun

Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz andIrene Stegun of the United States National Bureau of Standards (now the National Institute of Standards and Tech-nology). Its full title is Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.Since it was first published in 1964, the 1046 page Handbook has been one of the most comprehensive sourcesof information on special functions, containing definitions, identities, approximations, plots, and tables of values ofnumerous functions used in virtually all fields of applied mathematics.[1] The notation used in the Handbook is the defacto standard for much of applied mathematics today.At the time of its publication, the Handbook was an essential resource for practitioners. Nowadays, computer algebrasystems have replaced the function tables, but the Handbook remains an important reference source. The foreworddiscusses a meeting in 1954 in which it was agreed that “the advent of high-speed computing equipment changed thetask of table making but definitely did not remove the need for tables”.

More than 1,000 pages long, the Handbook of Mathematical Functions was first published in 1964and reprinted many times, with yet another reprint in 1999. Its influence on science and engineering isevidenced by its popularity. In fact, when New Scientist magazine recently asked some of the world’sleading scientists what single book they would want if stranded on a desert island, one distinguishedBritish physicist[2] said he would take the Handbook.

The Handbook is likely the most widely distributed and most cited NIST technical publication ofall time. Government sales exceed 150,000 copies, and an estimated three times as many have beenreprinted and sold by commercial publishers since 1965. During the mid-1990s, the book was citedevery 1.5 hours of each working day. And its influence will persist as it is currently being updated indigital format by NIST.

— NIST[3]

1.1 Editions

Because the Handbook is the work of U.S. federal government employees acting in their official capacity, it is notprotected by copyright. While it can be ordered from the Government Printing Office, it has also been reprintedby commercial publishers, most notably Dover Publications (ISBN 0-486-61272-4), and can be legally viewed anddownloaded off the web.

1.2 Successor

A digital successor to the Handbook, long under development at NIST, was released as the “Digital Library of Math-ematical Functions” (DLMF) on May 11, 2010, along with a printed version, the NIST Handbook of MathematicalFunctions, published by Cambridge University Press (ISBN 978-0-521-19225-5). More information can be found atNIST.

1

2 CHAPTER 1. ABRAMOWITZ AND STEGUN

1.3 See also• Numerical analysis

• Philip J. Davis, author of the Gamma function section and other sections of the book

• Digital Library of Mathematical Functions (DLMF), from the National Institute of Standards and Technology(NIST), is intended to be a replacement for Abramowitz and Stegun’s Handbook of Mathematical Functions

• Boole’s rule, a mathematical rule of integration sometimes known as Bode’s rule, due to a typo in Abramowitzand Stegun (1972, p. 886) that was subsequently propagated [4]

1.4 Notes[1] Boisvert, R. et al. (2011) A Special Functions Handbook for the Digital Age, NAMS 58(7), 905-911.

[2] Michael Berry, New Scientist 22 November 1997

[3] NIST at 100: Foundations for Progress, 1964:Mathematics Handbook Becomes Best Seller — http://www.100.nist.gov/ph_spaceage.htm

[4] “Boole’s Rule - from Wolfram MathWorld”. Mathworld.wolfram.com. 2009-10-27. Retrieved 2009-11-13.

1.5 References• Abramowitz, Milton; Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas,

Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0

• Boisvert, R. F.; Lozier, D. W. (2001), “Handbook of Mathematical Functions”, in Lide, D. R., A Century ofExcellence in Measurements Standards and Technology (PDF), CRC Press, pp. 135–139, ISBN 0-8493-1247-7A history of the activities leading up to and surrounding the development of the Handbook

1.6 External links• How to order the book from GPO.

• A high quality scan of the book, in PDF and TIFF formats, hosted at the University of Birmingham, UK

• The book in scanned format, hosted at Simon Fraser University.

• Another scanned version by ConvertIt.com

• Empanel version

• NIST Digital Library of Mathematical Functions, the digital companion to the Handbook

Chapter 2

Absolute value

For other uses, see Absolute value (disambiguation).In mathematics, the absolute value (ormodulus) |x| of a real number x is the non-negative value of x without regard

The absolute value of a number may be thought of as its distance from zero.

to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. Forexample, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may bethought of as its distance from zero.Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For examplean absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical andphysical contexts.

2.1 Terminology and notation

In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specifically for thecomplex absolute value,[1][2] and it was borrowed into English in 1866 as the Latin equivalent modulus.[1] The termabsolute value has been used in this sense from at least 1806 in French[3] and 1857 in English.[4] The notation |x|, witha vertical bar on each side, was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value includenumerical value[1] and magnitude.[1]

The same notation is used with sets to denote cardinality; the meaning depends on context.

2.2 Definition and properties

3

4 CHAPTER 2. ABSOLUTE VALUE

2.2.1 Real numbers

For any real number x the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of thequantity) and is defined as[6]

|x| =

{x, if x ≥ 0

−x, if x < 0.

As can be seen from the above definition, the absolute value of x is always either positive or zero, but never negative.From an analytic geometry point of view, the absolute value of a real number is that number’s distance from zeroalong the real number line, and more generally the absolute value of the difference of two real numbers is the distancebetween them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisationof the absolute value of the difference (see “Distance” below).Since the square root notation without sign represents the positive square root, it follows that

which is sometimes used as a definition of absolute value of real numbers.[7]

The absolute value has the following four fundamental properties:

Other important properties of the absolute value include:

Two other useful properties concerning inequalities are:

|a| ≤ b ⇐⇒ −b ≤ a ≤ b

|a| ≥ b ⇐⇒ a ≤ −b or b ≤ a

These relations may be used to solve inequalities involving absolute values. For example:

Absolute value is used to define the absolute difference, the standard metric on the real numbers.

2.2.2 Complex numbers

Since the complex numbers are not ordered, the definition given above for the real absolute value cannot be directlygeneralised for a complex number. However the geometric interpretation of the absolute value of a real number as itsdistance from 0 can be generalised. The absolute value of a complex number is defined as its distance in the complexplane from the origin using the Pythagorean theorem. More generally the absolute value of the difference of twocomplex numbers is equal to the distance between those two complex numbers.For any complex number

z = x+ iy,

where x and y are real numbers, the absolute value ormodulus of z is denoted |z| and is given by[8]

|z| =√x2 + y2.

2.2. DEFINITION AND PROPERTIES 5

Im

Re

y

−y

0 x

r

r

φ

φ

z=x+iy

z=x−iy

The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complexconjugate z have the same absolute value.

When the imaginary part y is zero this is the same as the absolute value of the real number x.When a complex number z is expressed in polar form as

6 CHAPTER 2. ABSOLUTE VALUE

z = reiθ

with r ≥ 0 and θ real, its absolute value is

|z| = r

The absolute value of a complex number can be written in the complex analogue of equation (1) above as:

|z| =√z · z

where z is the complex conjugate of z. Notice that, contrary to equation (1):

|z| ̸=√z2

The complex absolute value shares all the properties of the real absolute value given in equations (2)–(11) above.Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolutevalue as an endomorphism of the multiplicative group of the complex numbers.[9]

2.3 Absolute value function

1

2

3

4

−3 −2 −1 1 2 30

y = |x|

The graph of the absolute value function for real numbers

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It ismonotonically decreasing on the interval (−∞,0] and monotonically increasing on the interval [0,+∞). Since a realnumber and its opposite have the same absolute value, it is an even function, and is hence not invertible.Both the real and complex functions are idempotent.It is a piecewise linear, convex function.

2.3. ABSOLUTE VALUE FUNCTION 7

x

y

f (| x |)

| f (x ) |

f (x )

Composition of absolute value with a cubic function in different orders

2.3.1 Relationship to the sign function

The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum)function returns a number’s sign irrespective of its value. The following equations show the relationship between thesetwo functions:

|x| = x sgn(x),

8 CHAPTER 2. ABSOLUTE VALUE

or

|x| sgn(x) = x,

and for x ≠ 0,

sgn(x) = |x|x.

2.3.2 Derivative

The real absolute value function has a derivative for every x ≠ 0, but is not differentiable at x = 0. Its derivative for x≠ 0 is given by the step function[10][11]

d|x|dx

=x

|x|=

{−1 x < 0

1 x > 0.

The subdifferential of |x| at x = 0 is the interval [−1,1].[12]

The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violatesthe Cauchy–Riemann equations.[10]

The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalisedfunction, the second derivative may be taken as two times the Dirac delta function.

2.3.3 Antiderivative

The antiderivative (indefinite integral) of the absolute value function is

∫|x|dx =

x|x|2

+ C,

where C is an arbitrary constant of integration.

2.4 Distance

See also: Metric space

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complexnumber is the distance from that number to the origin, along the real number line, for real numbers, or in the complexplane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbersis the distance between them.The standard Euclidean distance between two points

a = (a1, a2, . . . , an)

and

b = (b1, b2, . . . , bn)

2.5. GENERALIZATIONS 9

in Euclidean n-space is defined as:

√√√√ n∑i=1

(ai − bi)2.

This can be seen to be a generalisation of |a − b|, since if a and b are real, then by equation (1),

|a− b| =√(a− b)2.

While if

a = a1 + ia2

and

b = b1 + ib2

are complex numbers, then

The above shows that the “absolute value” distance for the real numbers or the complex numbers, agrees with thestandard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclideanspaces respectively.The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity ofindiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion ofa distance function as follows:A real valued function d on a set X × X is called a metric (or a distance function) on X, if it satisfies the followingfour axioms:[13]

2.5 Generalizations

2.5.1 Ordered rings

The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if a is anelement of an ordered ring R, then the absolute value of a, denoted by |a|, is defined to be:[14]

|a| =

{a, if a ≥ 0

−a, if a ≤ 0

where −a is the additive inverse of a, and 0 is the additive identity element.

2.5.2 Fields

Main article: Absolute value (algebra)

10 CHAPTER 2. ABSOLUTE VALUE

The fundamental properties of the absolute value for real numbers given in (2)–(5) above, can be used to generalisethe notion of absolute value to an arbitrary field, as follows.A real-valued function v on a field F is called an absolute value (also a modulus, magnitude, value, or valuation)[15] ifit satisfies the following four axioms:

Where 0 denotes the additive identity element of F. It follows from positive-definiteness and multiplicativeness thatv(1) = 1, where 1 denotes the multiplicative identity element of F. The real and complex absolute values defined aboveare examples of absolute values for an arbitrary field.If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(a − b), is a metric and the followingare equivalent:

• d satisfies the ultrametric inequality d(x, y) ≤ max(d(x, z), d(y, z)) for all x, y, z in F.

•{v(∑n

k=11): n ∈ N

}is bounded in R.

• v(∑n

k=11)≤ 1 for every n ∈ N.

• v(a) ≤ 1 ⇒ v(1 + a) ≤ 1 for all a ∈ F.

• v(a+ b) ≤ max{v(a), v(b)} for all a, b ∈ F.

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwiseit is said to be Archimedean.[16]

2.5.3 Vector spaces

Main article: Norm (mathematics)

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, togeneralise the notion to an arbitrary vector space.A real-valued function on a vector space V over a field F, represented as ‖·‖, is called an absolute value, but moreusually a norm, if it satisfies the following axioms:For all a in F, and v, u in V,

The norm of a vector is also called its length or magnitude.In the case of Euclidean space Rn, the function defined by

∥(x1, x2, . . . , xn)∥ =

√√√√ n∑i=1

x2i

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector spaceR1, the absolute value is a norm, and is the p-norm (see Lp space) for any p. In fact the absolute value is the “only”norm on R1, in the sense that, for every norm ‖·‖ on R1, ‖x‖ = ‖1‖ ⋅ |x|. The complex absolute value is a special caseof the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified withthe Euclidean plane R2.

2.6. NOTES 11

2.6 Notes[1] Oxford English Dictionary, Draft Revision, June 2008

[2] Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877

[3] Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques prisdans l'espace, p. 105 at Google Books

[4] James Mill Peirce, A Text-book of Analytic Geometry at Google Books. The oldest citation in the 2nd edition of the OxfordEnglish Dictionary is from 1907. The term absolute value is also used in contrast to relative value.

[5] Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25

[6] Mendelson, p. 2.

[7] Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1., p. A5

[8] González, Mario O. (1992). Classical Complex Analysis. CRC Press. p. 19. ISBN 9780824784157.

[9] Lorenz, Falko (2008), Algebra. Vol. II. Fields with structure, algebras and advanced topics, Universitext, New York:Springer, p. 39, doi:10.1007/978-0-387-72488-1, ISBN 978-0-387-72487-4, MR 2371763.

[10] Weisstein, Eric W. Absolute Value. From MathWorld – A Wolfram Web Resource.

[11] Bartel and Sherbert, p. 163

[12] Peter Wriggers, Panagiotis Panatiotopoulos, eds., New Developments in Contact Problems, 1999, ISBN 3-211-83154-1, p.31–32

[13] These axioms are not minimal; for instance, non-negativity can be derived from the other three: 0 = d(a, a) ≤ d(a, b) +d(b, a) = 2d(a, b).

[14] Mac Lane, p. 264.

[15] Shechter, p. 260. This meaning of valuation is rare. Usually, a valuation is the logarithm of the inverse of an absolutevalue

[16] Shechter, pp. 260–261.

2.7 References• Bartle; Sherbert; Introduction to real analysis (4th ed.), John Wiley & Sons, 2011 ISBN 978-0-471-43331-6.

• Nahin, Paul J.; An Imaginary Tale; Princeton University Press; (hardcover, 1998). ISBN 0-691-02795-1.

• Mac Lane, Saunders, Garrett Birkhoff, Algebra, AmericanMathematical Soc., 1999. ISBN 978-0-8218-1646-2.

• Mendelson, Elliott, Schaum’s Outline of Beginning Calculus, McGraw-Hill Professional, 2008. ISBN 978-0-07-148754-2.

• O'Connor, J.J. and Robertson, E.F.; “Jean Robert Argand”.

• Schechter, Eric; Handbook of Analysis and Its Foundations, pp. 259–263, “Absolute Values”, Academic Press(1997) ISBN 0-12-622760-8.

2.8 External links• Hazewinkel, Michiel, ed. (2001), “Absolute value”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• absolute value at PlanetMath.org.

• Weisstein, Eric W., “Absolute Value”, MathWorld.

Chapter 3

Ackermann function

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest[1] andearliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursivefunctions are total and computable, but the Ackermann function illustrates that not all total computable functions areprimitive recursive.After Ackermann’s publication[2] of his function (which had three nonnegative integer arguments), many authorsmodified it to suit various purposes, so that today “the Ackermann function” may refer to any of numerous variantsof the original function. One common version, the two-argument Ackermann–Péter function, is defined as followsfor nonnegative integers m and n:

A(m,n) =

n+ 1 ifm = 0

A(m− 1, 1) ifm > 0 and n = 0

A(m− 1, A(m,n− 1)) ifm > 0 and n > 0.

Its value grows rapidly, even for small inputs. For example A(4,2) is an integer of 19,729 decimal digits.[3]

3.1 History

In the late 1920s, the mathematicians Gabriel Sudan andWilhelm Ackermann, students of David Hilbert, were study-ing the foundations of computation. Both Sudan and Ackermann are credited[4] with discovering total computablefunctions (termed simply “recursive” in some references) that are not primitive recursive. Sudan published the lesser-known Sudan function, then shortly afterwards and independently, in 1928, Ackermann published his function φ .Ackermann’s three-argument function, φ(m,n, p) , is defined such that for p = 0, 1, 2, it reproduces the basic oper-ations of addition, multiplication, and exponentiation as

φ(m,n, 0) = m+ n,

φ(m,n, 1) = m · n,

φ(m,n, 2) = mn,

and for p > 2 it extends these basic operations in a way that can be compared to the hyperoperations: (Aside from itshistoric role as a total-computable-but-not-primitive-recursive function, Ackermann’s original function is seen to ex-tend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann’sfunction that are specifically designed for that purpose—such as Goodstein’s hyperoperation sequence.)In On the Infinite, David Hilbert hypothesized that the Ackermann function was not primitive recursive, but it wasAckermann, Hilbert’s personal secretary and former student, who actually proved the hypothesis in his paper OnHilbert’s Construction of the Real Numbers. On the Infinite was Hilbert’s most important paper on the foundationsof mathematics, serving as the heart of Hilbert’s program to secure the foundation of transfinite numbers by basingthem on finite methods.[2][5]

12

3.2. DEFINITION AND PROPERTIES 13

Rózsa Péter and Raphael Robinson later developed a two-variable version of the Ackermann function that becamepreferred by many authors.[6]

3.2 Definition and properties

Ackermann’s original three-argument function φ(m,n, p) is defined recursively as follows for nonnegative integersm, n, and p:

φ(m,n, p) =

φ(m,n, 0) = m+ n

φ(m, 0, 1) = 0

φ(m, 0, 2) = 1

φ(m, 0, p) = m for p > 2

φ(m,n, p) = φ(m,φ(m,n− 1, p), p− 1) for n > 0 and p > 0.

Of the various two-argument versions, the one developed by Péter and Robinson (called “the” Ackermann functionby some authors) is defined for nonnegative integers m and n as follows:

A(m,n) =

n+ 1 ifm = 0

A(m− 1, 1) ifm > 0 and n = 0

A(m− 1, A(m,n− 1)) ifm > 0 and n > 0.

It may not be immediately obvious that the evaluation of A(m,n) always terminates. However, the recursion isbounded because in each recursive application eitherm decreases, orm remains the same and n decreases. Each timethat n reaches zero, m decreases, so m eventually reaches zero as well. (Expressed more technically, in each case thepair (m, n) decreases in the lexicographic order on pairs, which is a well-ordering, just like the ordering of singlenon-negative integers; this means one cannot go down in the ordering infinitely many times in succession.) However,when m decreases there is no upper bound on how much n can increase—and it will often increase greatly.The Péter-Ackermann function can also be expressed in terms of various other versions of the Ackermann function:

• the indexed version of Knuth’s up-arrow notation (extended to integer indices ≥ −2):

A(m,n) = 2 ↑m−2 (n+ 3)− 3.

The part of the definition A(m, 0) = A(m−1, 1) corresponds to 2 ↑m+1 3 = 2 ↑m 4.

• hyper operators:

A(m,n) = 2[m](n+ 3)− 3

• Conway chained arrow notation:

A(m,n) = (2 → (n+ 3) → (m− 2))− 3 form ≥ 3

hence

2 → n → m = A(m+ 2, n− 3) + 3 for n > 2 .

(n=1 and n=2 would correspond with A(m,−2) = −1 and A(m,−1) = 1, which could logically be added.)

14 CHAPTER 3. ACKERMANN FUNCTION

For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n (at mostexponentially). For m ≥ 4, however, it grows much more quickly; even A(4, 2) is about 2×1019728, and the decimalexpansion of A(4, 3) is very large by any typical measure.Logician Harvey Friedman defines a version of the Ackermann function as follows:

• For n = 0: A(m, n) = 1

• For m = 1: A(m, n) = 2n

• Else: A(m, n) = A(m - 1, A(m, n - 1))

He also defines a single-argument version A(n) = A(n, n).[7]

A single-argument version A(k) = A(k, k) that increases both m and n at the same time dwarfs every primitiverecursive function, including very fast-growing functions such as the exponential function, the factorial function,multi- and superfactorial functions, and even functions defined using Knuth’s up-arrow notation (except when theindexed up-arrow is used). It can be seen that A(n) is roughly comparable to fω(n) in the fast-growing hierarchy.This extreme growth can be exploited to show that f, which is obviously computable on a machine with infinitememory such as a Turing machine and so is a computable function, grows faster than any primitive recursive functionand is therefore not primitive recursive. In a category with exponentials, using the isomorphism A × B → C ∼=A → (B → C) (in computer science, this is called currying), the Ackermann function may be defined via primitiverecursion over higher-order functionals as follows:

Ack(0) = SuccAck(m+ 1) = Iter(Ack(m))

where Succ is the usual successor function and Iter is defined by primitive recursion as well:

Iter(f)(0) = f(1)Iter(f)(n+ 1) = f(Iter(f)(n)).One interesting aspect of the Ackermann function is that the only arithmetic operations it ever uses are addition andsubtraction of 1. Its properties come solely from the power of unlimited recursion. This also implies that its runningtime is at least proportional to its output, and so is also extremely huge. In actuality, for most cases the running timeis far larger than the output; see below.

3.3 Table of values

Computing the Ackermann function can be restated in terms of an infinite table. We place the natural numbers alongthe top row. To determine a number in the table, take the number immediately to the left, then look up the requirednumber in the previous row, at the position given by the number just taken. If there is no number to its left, simplylook at the column headed “1” in the previous row. Here is a small upper-left portion of the table:The numbers here which are only expressed with recursive exponentiation or Knuth arrows are very large and wouldtake up too much space to notate in plain decimal digits.Despite the large values occurring in this early section of the table, some even larger numbers have been defined, suchas Graham’s number, which cannot be written with any small number of Knuth arrows. This number is constructedwith a technique similar to applying the Ackermann function to itself recursively.This is a repeat of the above table, but with the values replaced by the relevant expression from the function definitionto show the pattern clearly:

3.4 Expansion

To see how the Ackermann function grows so quickly, it helps to expand out some simple expressions using the rulesin the original definition. For example, we can fully evaluate A(1, 2) in the following way:

3.5. INVERSE 15

A(1, 2) = A(0, A(1, 1))

= A(0, A(0, A(1, 0)))

= A(0, A(0, A(0, 1)))

= A(0, A(0, 2))

= A(0, 3)

= 4.

To demonstrate how A(4, 3) 's computation results in many steps and in a large number:

A(4, 3) = A(3, A(4, 2))

= A(3, A(3, A(4, 1)))

= A(3, A(3, A(3, A(4, 0))))

= A(3, A(3, A(3, A(3, 1))))

= A(3, A(3, A(3, A(2, A(3, 0)))))

= A(3, A(3, A(3, A(2, A(2, 1)))))

= A(3, A(3, A(3, A(2, A(1, A(2, 0))))))

= A(3, A(3, A(3, A(2, A(1, A(1, 1))))))

= A(3, A(3, A(3, A(2, A(1, A(0, A(1, 0)))))))

= A(3, A(3, A(3, A(2, A(1, A(0, A(0, 1)))))))

= A(3, A(3, A(3, A(2, A(1, A(0, 2))))))

= A(3, A(3, A(3, A(2, A(1, 3)))))

= A(3, A(3, A(3, A(2, A(0, A(1, 2))))))

= A(3, A(3, A(3, A(2, A(0, A(0, A(1, 1)))))))

= A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(1, 0))))))))

= A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(0, 1))))))))

= A(3, A(3, A(3, A(2, A(0, A(0, A(0, 2)))))))

= A(3, A(3, A(3, A(2, A(0, A(0, 3))))))

= A(3, A(3, A(3, A(2, A(0, 4)))))

= A(3, A(3, A(3, A(2, 5))))

= . . .

= A(3, A(3, A(3, 13)))

= . . .

= A(3, A(3, 65533))

= . . .

= A(3, 265536 − 3)

= . . .

= 2265536

− 3.

Written as a power of 10, this is roughly equivalent to 106.031×1019727 .

3.5 Inverse

Since the function f (n) = A(n, n) considered above grows very rapidly, its inverse function, f−1, grows very slowly.This inverse Ackermann function f−1 is usually denoted by α. In fact, α(n) is less than 5 for any practical input sizen, since A(4, 4) is on the order of 222

216

.

16 CHAPTER 3. ACKERMANN FUNCTION

This inverse appears in the time complexity of some algorithms, such as the disjoint-set data structure and Chazelle'salgorithm for minimum spanning trees. Sometimes Ackermann’s original function or other variations are used inthese settings, but they all grow at similarly high rates. In particular, some modified functions simplify the expressionby eliminating the −3 and similar terms.A two-parameter variation of the inverse Ackermann function can be defined as follows, where ⌊x⌋ is the floorfunction:

α(m,n) = min{i ≥ 1 : A(i, ⌊m/n⌋) ≥ log2 n}.

This function arises in more precise analyses of the algorithms mentioned above, and gives a more refined time bound.In the disjoint-set data structure,m represents the number of operations while n represents the number of elements; inthe minimum spanning tree algorithm, m represents the number of edges while n represents the number of vertices.Several slightly different definitions of α(m, n) exist; for example, log2 n is sometimes replaced by n, and the floorfunction is sometimes replaced by a ceiling.Other studies might define an inverse function of one where m is set to a constant, such that the inverse applies to aparticular row.[8]

3.6 Use as benchmark

The Ackermann function, due to its definition in terms of extremely deep recursion, can be used as a benchmark of acompiler's ability to optimize recursion. The first use of Ackermann’s function in this way was by Yngve Sundblad,The Ackermann function. A Theoretical, computational and formula manipulative study.[9]

This seminal paper was taken up by Brian Wichmann (co-author of the Whetstone benchmark) in a trilogy of paperswritten between 1975 and 1982.[10][11][12]

3.7 Ackermann numbers

In The Book of Numbers, John Horton Conway and Richard K. Guy define the sequence of Ackermann numbers tobe 0[0]0, 1[1]1, 2[2]2, 3[3]3, etc.;[13] that is, the n-th Ackermann number is defined to be n[n]n (n = 0, 1, 2, 3, ...),where a[n]b is the hyperoperation.The first few Ackermann numbers are (sequence A189896 in OEIS)

• 0[0]0 = 0 + 1 = 1• 1[1]1 = 1 + 1 = 2,• 2[2]2 = 2 · 2 = 4,• 3[3]3 = 33 = 27,• 4[4]4 = 4[3]4[3]4[3]4 = 444

4

= 44256 = 42512 = 22513 ,

• 5[5]5 = 5[4]5[4]5[4]5[4]5 = 5[4]5[4]5[4](5[3]5[3]5[3]5[3]5) = 5[4]5[4]5[4](55555

)

The sixth Ackermann number, 6[6]6, can be written in terms of tetration towers as follows:

6[6]6 = 6[5]6[5]6[5]6[5]6[5]6 = 6[5]6[5]6[5]6[5](6[4]6[4]6[4]6[4]6[4]6)

Alternatively, this can be written in terms of exponentiation towers as

6[6]6 = 6[5](6[5](6[5](6[5](6[5]6)))) =

66····6}

66···6}. . . 66

66}6,

3.8. SEE ALSO 17

where the number of towers on the previous line (including the rightmost “6”) is

66····6}

66···6}. . . 66

66}6,

where the number of towers on the previous line (including the rightmost “6”) is

66···6}66

···6}66

66}6,

where the number of towers on the previous line (including the rightmost “6”) is

66···6}66

···6}66

66}6,

where the number of towers on the previous line (including the rightmost “6”) is

66···6}66

···6}66

66}6,

where the number of 6s in each tower, on each of the lines above, is specified by the value of the next tower to itsright (as indicated by a brace).The above three lines of exponentiation towers correspond to the indicated three applications of6[5]n = 6[4](6[4](6[4](6[4](6[4]6))))︸ ︷︷ ︸

n 6s

.

3.8 See also• Computability theory

• Double recursion

• Fast-growing hierarchy

• Goodstein function

• Primitive recursive function

• Recursion (computer science)

3.9 References[1] Monin, Jean-Francois; Hinchey, M. G. (2003), Understanding Formal Methods, Springer, p. 61, ISBN 9781852332471,

There are total functions that cannot be defined by a primitive recursive presentation, but they are not that easy to find. Oneof the simplest is the Ackermann function.

[2] Wilhelm Ackermann (1928). “Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen 99: 118–133.doi:10.1007/BF01459088.

[3] Decimal expansion of A(4,2) Archived March 17, 2008 at the Wayback Machine

[4] Cristian Calude, Solomon Marcus and Ionel Tevy (November 1979). “The first example of a recursive function which isnot primitive recursive”. Historia Math. 6 (4): 380–84. doi:10.1016/0315-0860(79)90024-7.

[5] von Heijenoort. From Frege To Gödel, 1967.

[6] Raphael M. Robinson (1948). “Recursion and Double Recursion”. Bulletin of the American Mathematical Society 54 (10):987–93. doi:10.1090/S0002-9904-1948-09121-2.

[7] http://www.math.osu.edu/~{}friedman.8/pdf/AckAlgGeom102100.pdf

[8] An inverse-Ackermann style lower bound for the online minimum spanning tree verification problem November 2002

[9] Sundblad, Yngve (1971-03-01). “The Ackermann function. a theoretical, computational, and formula manipulative study”.BIT Numerical Mathematics (Kluwer Academic Publishers) 11 (1): 107–119. doi:10.1007/BF01935330.

[10] “Ackermann’s Function: A Study In The Efficiency Of Calling Procedures” (PDF). 1975.

18 CHAPTER 3. ACKERMANN FUNCTION

[11] “How to Call Procedures, or Second Thoughts on Ackermann’s Function” (PDF). 1977.

[12] “Latest results from the procedure calling test, Ackermann’s function” (PDF). 1982.

[13] John Horton Conway and Richard K. Guy. The Book of Numbers. New York: Springer-Verlag, pp. 60-61, 1996. ISBN978-0-387-97993-9

3.10 External links• Hazewinkel, Michiel, ed. (2001), “Ackermann function”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Ackermann function”, MathWorld.

• Black, Paul E. “Ackermann’s function”. Dictionary of Algorithms and Data Structures. NIST.

• An animated Ackermann function calculator

• Scott Aaronson,Who can name the biggest number? (1999)

• Ackermann function’s. Includes a table of some values.

• Hyper-operations: Ackermann’s Function and New Arithmetical Operation

• Robert Munafo’s Large Numbers describes several variations on the definition of A.

• Gabriel Nivasch, Inverse Ackermann without pain on the inverse Ackermann function.

• Raimund Seidel, Understanding the inverse Ackermann function (PDF presentation).

• The Ackermann function written in different programming languages, (on Rosetta Code)

• Ackermann’s Function (Archived 2009-10-24)—Some study and programming by Harry J. Smith.

Chapter 4

Airy function

This article is about the Airy special function. For the Airy stress function employed in solid mechanics, see Stressfunctions.

In the physical sciences, the Airy function Ai(x) is a special function named after the British astronomer GeorgeBiddell Airy (1801–92). The function Ai(x) and the related function Bi(x), which is also called the Airy function,but sometimes referred to as the Bairy function, are solutions to the differential equation

d2y

dx2− xy = 0,

known as the Airy equation or the Stokes equation. This is the simplest second-order linear differential equationwith a turning point (a point where the character of the solutions changes from oscillatory to exponential).The Airy function is the solution to Schrödinger’s equation for a particle confined within a triangular potential welland for a particle in a one-dimensional constant force field. For the same reason, it also serves to provide uniformsemiclassical approximations near a turning point in the WKB approximation, when the potential may be locallyapproximated by a linear function of position. The triangular potential well solution is directly relevant for the un-derstanding of many semiconductor devices.The Airy function also underlies the form of the intensity near an optical directional caustic, such as that of therainbow. Historically, this was the mathematical problem that led Airy to develop this special function. The Airyfunction is also important in microscopy and astronomy; it describes the pattern, due to diffraction and interference,produced by a point source of light (one of which is smaller than the resolution limit of a microscope or telescope).

4.1 Definitions

For real values of x, the Airy function of the first kind can be defined by the improper Riemann integral:

Ai(x) = 1

π

∫ ∞

0

cos(

t3

3 + xt)dt ≡ 1

πlimb→∞

∫ b

0

cos(

t3

3 + xt)dt,

which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out (ascan be checked by integration by parts).y = Ai(x) satisfies the Airy equation

y′′ − xy = 0.

This equation has two linearly independent solutions. Up to scalar multiplication, Ai(x) is the solution subject to thecondition y→0 as x→∞. The standard choice for the other solution is the Airy function of the second kind, denoted

19

20 CHAPTER 4. AIRY FUNCTION

Ai(x)Bi(x)

0.50

0.25

0.00

−0.25

−0.50

−15 −10 −5 0 5

x

0.75

1.00

Plot of Ai(x) in red and Bi(x) in blue

Bi(x). It is defined as the solution with the same amplitude of oscillation as Ai(x) as x → −∞ which differs in phaseby π/2:

Bi(x) = 1

π

∫ ∞

0

[exp

(− t3

3 + xt)+ sin

(t3

3 + xt) ]

dt.

4.2 Properties

The values of Ai(x) and Bi(x) and their derivatives at x = 0 are given by

Ai(0) = 1

323Γ( 23 )

, Ai′(0) = − 1

313Γ( 13 )

,

Bi(0) = 1

316Γ( 23 )

, Bi′(0) = 316

Γ(13 ).

Here, Γ denotes the Gamma function. It follows that the Wronskian of Ai(x) and Bi(x) is 1/π.When x is positive, Ai(x) is positive, convex, and decreasing exponentially to zero, while Bi(x) is positive, convex, andincreasing exponentially. When x is negative, Ai(x) and Bi(x) oscillate around zero with ever-increasing frequencyand ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.The Airy functions are orthogonal[1] in the sense that

∫ ∞

−∞Ai(t+ x)Ai(t+ y)dt = δ(x− y)

again using an improper Riemann integral.

4.3. ASYMPTOTIC FORMULAE 21

4.3 Asymptotic formulae

Ai(blue) and sinusoidal/exponential asymptotic form of Ai(magenta)

Bi(blue) and sinusoidal/exponential asymptotic form of Bi(magenta)

As explained below, the Airy functions can be extended to the complex plane, giving entire functions. The asymptotic

22 CHAPTER 4. AIRY FUNCTION

behaviour of the Airy functions as |z| goes to infinity at a constant value of arg(z) depends on arg(z): this is called theStokes phenomenon. For |arg(z)| < π we have the following asymptotic formula for Ai(z):[2]

Ai(z) ∼ e−23 z

32

2√π z

14

[ ∞∑n=0

(−1)nΓ(n+ 56 )Γ(n+ 1

6 )(34

)n2πn!z3n/2

].

and a similar one for Bi(z), but only applicable when |arg(z)| < π/3:

Bi(z) ∼ e23 z

32

√π z

14

[ ∞∑n=0

Γ(n+ 56 )Γ(n+ 1

6 )(34

)n2πn!z3n/2

].

A more accurate formula for Ai(z) and a formula for Bi(z) when π/3 < |arg(z)| < π or, equivalently, for Ai(−z) andBi(−z) when |arg(z)| < 2π/3 but not zero, are:[3]

Ai(−z) ∼sin

(23z

32 + π

4

)√π z

14

Bi(−z) ∼cos

(23z

32 + π

4

)√π z

14

.

When |arg(z)| = 0 these are good approximations but are not asymptotic because the ratio between Ai(−z) or Bi(−z)and the above approximation goes to infinity whenever the sine or cosine goes to zero. Asymptotic expansions forthese limits are also available. These are listed in (Abramowitz and Stegun, 1954) and (Olver, 1974).

4.4 Complex arguments

We can extend the definition of the Airy function to the complex plane by

Ai(z) = 1

2πi

∫C

exp(

t3

3 − zt)dt,

where the integral is over a path C starting at the point at infinity with argument −π/2 and ending at the point atinfinity with argument π/2. Alternatively, we can use the differential equation y′′ − xy = 0 to extend Ai(x) and Bi(x)to entire functions on the complex plane.The asymptotic formula for Ai(x) is still valid in the complex plane if the principal value of x2/3 is taken and x isbounded away from the negative real axis. The formula for Bi(x) is valid provided x is in the sector {x ∈ C : |arg(x)|< (π/3)−δ} for some positive δ. Finally, the formulae for Ai(−x) and Bi(−x) are valid if x is in the sector {x ∈ C :|arg(x)| < (2π/3)−δ}.It follows from the asymptotic behaviour of the Airy functions that both Ai(x) and Bi(x) have an infinity of zeros onthe negative real axis. The function Ai(x) has no other zeros in the complex plane, while the function Bi(x) also hasinfinitely many zeros in the sector {z ∈ C : π/3 < |arg(z)| < π/2}.

4.4.1 Plots

4.5 Relation to other special functions

For positive arguments, the Airy functions are related to the modified Bessel functions:

4.6. FOURIER TRANSFORM 23

Ai(x) = 1

π

√x

3K 1

3

(23x

32

),

Bi(x) =√

x

3

(I 1

3

(23x

32

)+ I− 1

3

(23x

32

)).

Here, I±₁/₃ and K₁/₃ are solutions of

x2y′′ + xy′ −(x2 + 1

9

)y = 0.

The first derivative of Airy function is

Ai′(x) = − x

π√3K 2

3

(23x

32

).

Functions K₁/₃ and K₂/₃ can be represented in terms of rapidly converged integrals[4] (see also modified Bessel func-tions )For negative arguments, the Airy function are related to the Bessel functions:

Ai(−x) =

√x

9

(J 1

3

(23x

32

)+ J− 1

3

(23x

32

)),

Bi(−x) =

√x

3

(J− 1

3

(23x

32

)− J 1

3

(23x

32

)).

Here, J±₁/₃ are solutions of

x2y′′ + xy′ +(x2 − 1

9

)y = 0.

The Scorer’s functions solve the equation y′′ − xy = 1/π. They can also be expressed in terms of the Airy functions:

Gi(x) = Bi(x)∫ ∞

x

Ai(t) dt+ Ai(x)∫ x

0

Bi(t) dt,

Hi(x) = Bi(x)∫ x

−∞Ai(t) dt− Ai(x)

∫ x

−∞Bi(t) dt.

4.6 Fourier transform

Using the definition of the Airy function Ai(x), it is straightforward to show its Fourier transform is given by

F(Ai)(k) :=∫ ∞

−∞Ai(x) e−2πikx dx = e

i3 (2πk)

3

.

4.7 Fabry–Pérot interferometer Airy Function

The transmittance function of a Fabry–Pérot interferometer is also referred to as the Airy Function:[5]

Te =1

1 + F sin2( δ2 ),

24 CHAPTER 4. AIRY FUNCTION

where both surfaces have reflectance R and

F =4R

(1−R)2

is the coefficient of finesse.

4.8 History

The Airy function is named after the British astronomer and physicist George Biddell Airy (1801–1892), who en-countered it in his early study of optics in physics (Airy 1838). The notation Ai(x) was introduced by Harold Jeffreys.Airy had become the British Astronomer Royal in 1835, and he held that post until his retirement in 1881.

4.9 See also

• The proof of Witten’s conjecture used a matrix-valued generalization of the Airy function.

• Airy zeta function

4.10 Notes

[1] David E. Aspnes, Physical Review, 147, 554 (1966)

[2] Abramowitz & Stegun (1970, p. 448), Eqns 10.4.59 and 10.4.63

[3] Abramowitz & Stegun (1970, p. 448), Eqns 10.4.60 and 10.4.64

[4] M.Kh.Khokonov. Cascade Processes of Energy Loss by Emission of Hard Photons // JETP, V.99, No.4, pp. 690-707 \(2004).

[5] Hecht, Eugene (1987). Optics (2nd ed. ed.). Addison Wesley. ISBN 0-201-11609-X. Sect. 9.6

4.11 References

• Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 10”, Handbook of Mathematical Functionswith Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 446, ISBN 978-0486612720, MR0167642.

• Airy (1838), “On the intensity of light in the neighbourhood of a caustic”, Transactions of the CambridgePhilosophical Society (University Press) 6: 379–402

• Olver (1974). Asymptotics and Special Functions, Chapter 11. Academic Press, New York.

• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), “Section 6.6.3. Airy Functions”, NumericalRecipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

• Vallée, Olivier; Soares, Manuel (2004), Airy functions and applications to physics, London: Imperial CollegePress, ISBN 978-1-86094-478-9, MR 2114198

4.12. EXTERNAL LINKS 25

4.12 External links• Hazewinkel, Michiel, ed. (2001), “Airy functions”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Airy Functions”, MathWorld.

• Wolfram function pages for Ai and Bi functions. Includes formulas, function evaluator, and plotting calculator.

• Olver, F. W. J. (2010), “Airy and related functions”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert,Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN978-0521192255, MR 2723248

Chapter 5

Algebraic function

This article is about algebraic functions in calculus, mathematical analysis, and abstract algebra. For functions inelementary algebra, see function (mathematics).

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quiteoften algebraic functions can be expressed using a finite number of terms, involving only the algebraic operationsaddition, subtraction, multiplication, division, and raising to a fractional power:

f(x) = 1/x, f(x) =√x, f(x) =

√1 + x3

x3/7 −√7x1/3

are typical examples.However, some algebraic functions cannot be expressed by such finite expressions (as proven by Galois and NielsAbel), as it is for example the case of the function defined by

f(x)5 + f(x)4 + x = 0

In more precise terms, an algebraic function of degree n in one variable x is a function y = f(x) that satisfies apolynomial equation

an(x)yn + an−1(x)y

n−1 + · · ·+ a0(x) = 0

where the coefficients ai(x) are polynomial functions of x, with coefficients belonging to a set S. Quite often, S = Q ,and one then talks about “function algebraic overQ ", and the evaluation at a given rational value of such an algebraicfunction gives an algebraic number.A functionwhich is not algebraic is called a transcendental function, as it is for example the case of exp(x), tan(x), ln(x),Γ(x). A composition of transcendental functions can give an algebraic function: f(x) = cos(arcsin(x)) =

√1− x2 .

As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, but nfunctions, sometimes also called branches. Consider for example the equation of the unit circle: y2 + x2 = 1. Thisdetermines y, except only up to an overall sign; accordingly, it has two branches: y = ±

√1− x2.

An algebraic function in m variables is similarly defined as a function y which solves a polynomial equation in m +1 variables:

p(y, x1, x2, . . . , xm) = 0.

It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is thenguaranteed by the implicit function theorem.Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field ofrational functions K(x1,...,xm).

26

5.1. ALGEBRAIC FUNCTIONS IN ONE VARIABLE 27

5.1 Algebraic functions in one variable

5.1.1 Introduction and overview

The informal definition of an algebraic function provides a number of clues about the properties of algebraic func-tions. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can beformed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. Of course, this issomething of an oversimplification; because of casus irreducibilis (and more generally the fundamental theorem ofGalois theory), algebraic functions need not be expressible by radicals.First, note that any polynomial function y = p(x) is an algebraic function, since it is simply the solution y to theequation

y − p(x) = 0.

More generally, any rational function y = p(x)q(x) is algebraic, being the solution to

q(x)y − p(x) = 0.

Moreover, the nth root of any polynomial y = n√p(x) is an algebraic function, solving the equation

yn − p(x) = 0.

Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solutionto

an(x)yn + · · ·+ a0(x) = 0,

for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of xand y and gathering terms,

bm(y)xm + bm−1(y)xm−1 + · · ·+ b0(y) = 0.

Writing x as a function of y gives the inverse function, also an algebraic function.However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one.The inverse is the algebraic “function” x = ±√

y . Another way to understand this, is that the set of branches of thepolynomial equation defining our algebraic function is the graph of an algebraic curve.

5.1.2 The role of complex numbers

From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. Firstof all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence anypolynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions notexceeding the degree of p in x) for y at each point x, provided we allow y to assume complex as well as real values.Thus, problems to do with the domain of an algebraic function can safely be minimized.Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express thefunction in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (seecasus irreducibilis). For example, consider the algebraic function determined by the equation

y3 − xy + 1 = 0.

28 CHAPTER 5. ALGEBRAIC FUNCTION

A graph of three branches of the algebraic function y, where y3 − xy + 1 = 0, over the domain 3/22/3 < x < 50.

Using the cubic formula, we get

y = − 2x3√

−108 + 12√81− 12x3

+3√

−108 + 12√81− 12x3

6.

For x ≤ 33√4

, the square root is real and the cubic root is thus well defined, providing the unique real root. On theother hand, for x > 3

3√4, the square root is not real, and one has to choose, for the square root, either non real-square

root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the twoterms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanyingimage.It may be proven that there is no way to express this function in terms nth roots using real numbers only, even thoughthe resulting function is real-valued on the domain of the graph shown.On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of complexanalysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraicfunction is in fact an analytic function, at least in the multiple-valued sense.Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x0 ∈ C is such that thepolynomial p(x0,y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhoodof x0. Choose a system of n non-overlapping discs Δi containing each of these zeros. Then by the argument principle

1

2πi

∮∂∆i

py(x0, y)

p(x0, y)dy = 1.

By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x,y) has only one root in Δi, given bythe residue theorem:

fi(x) =1

2πi

∮∂∆i

ypy(x, y)

p(x, y)dy

which is an analytic function.

5.2. HISTORY 29

5.1.3 Monodromy

Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi(x),provided that x is not a critical point of p(x, y). A critical point is a point where the number of distinct zeros is smallerthan the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminantvanishes. Hence there are only finitely many such points c1, ..., cm.A close analysis of the properties of the function elements fi near the critical points can be used to show that themonodromy cover is ramified over the critical points (and possibly the point at infinity). Thus the entire functionassociated to the fi has at worst algebraic poles and ordinary algebraic branchings over the critical points.Note that, away from the critical points, we have

p(x, y) = an(x)(y − f1(x))(y − f2(x)) · · · (y − fn(x))

since the fi are by definition the distinct zeros of p. The monodromy group acts by permuting the factors, and thusforms the monodromy representation of the Galois group of p. (The monodromy action on the universal coveringspace is related but different notion in the theory of Riemann surfaces.)

5.2 History

The ideas surrounding algebraic functions go back at least as far as René Descartes. The first discussion of algebraicfunctions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in whichhe writes:

let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methodsof division and extraction of roots, reduce it into an infinite series ascending or descending according tothe dimensions of x, and then find the integral of each of the resulting terms.

5.3 See also• Algebraic expression

• Analytic function

• Complex function

• Elementary function

• Function (mathematics)

• Generalized function

• List of special functions and eponyms

• List of types of functions

• Polynomial

• Rational function

• Special functions

• Transcendental function

5.4 References• Ahlfors, Lars (1979). Complex Analysis. McGraw Hill.

• van der Waerden, B.L. (1931). Modern Algebra, Volume II. Springer.

30 CHAPTER 5. ALGEBRAIC FUNCTION

5.5 External links• Definition of “Algebraic function” in the Encyclopedia of Math

• Weisstein, Eric W., “Algebraic Function”, MathWorld.

• Algebraic Function at PlanetMath.org.

• Definition of “Algebraic function” in David J. Darling's Internet Encyclopedia of Science

Chapter 6

Anger function

In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as

Jν(z) =1

π

∫ π

0

cos(νθ − z sin θ) dθ

and is closely related to Bessel functions.The Weber function (also known as Lommel-Weber function), introduced by H. F. Weber (1879), is a closelyrelated function defined by

Eν(z) =1

π

∫ π

0

sin(νθ − z sin θ) dθ

and is closely related to Bessel functions of the second kind.

6.1 Relation between Weber and Anger functions

The Anger and Weber functions are related by

sin(πν)Jν(z) = cos(πν)Eν(z)− E−ν(z)

− sin(πν)Eν(z) = cos(πν)Jν(z)− J−ν(z)

so in particular if ν is not an integer they can be expressed as linear combinations of each other. If ν is an integerthen Anger functions Jν are the same as Bessel functions Jν, and Weber functions can be expressed as finite linearcombinations of Struve functions.

6.2 Differential equations

TheAnger andWeber functions are solutions of inhomogeneous forms of Bessel’s equation z2y′′+zy′+(z2−ν2)y =0 . More precisely, the Anger functions satisfy the equation

z2y′′ + zy′ + (z2 − ν2)y = (z − ν) sin(πz)/π

and the Weber functions satisfy the equation

z2y′′ + zy′ + (z2 − ν2)y = −((z + ν) + (z − ν) cos(πz))/π.

31

32 CHAPTER 6. ANGER FUNCTION

6.3 References• Abramowitz, Milton; Stegun, Irene A., eds. (1965), “Chapter 12”, Handbook of Mathematical Functions

with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 498, ISBN 978-0486612720, MR0167642.

• C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29

• Paris, R. B. (2010), “Anger-Weber Functions”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, RonaldF.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248

• Prudnikov, A.P. (2001), “Anger function”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Prudnikov, A.P. (2001), “Weber function”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• G.N. Watson, “A treatise on the theory of Bessel functions”, 1–2, Cambridge Univ. Press (1952)

• H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76

Chapter 7

Arithmetic–geometric mean

In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers x and y is defined as follows:First compute the arithmetic mean of x and y and call it a1. Next compute the geometric mean of x and y and call itg1; this is the square root of the product xy:

a1 = 12 (x+ y)

g1 =√xy

Then iterate this operation with a1 taking the place of x and g1 taking the place of y. In this way, two sequences (an)and (gn) are defined:

an+1 = 12 (an + gn)

gn+1 =√angn

These two sequences converge to the same number, which is the arithmetic–geometricmean of x and y; it is denotedby M(x, y), or sometimes by agm(x, y).This can be used for algorithmic purposes as in the AGM method.

7.1 Example

To find the arithmetic–geometric mean of a0 = 24 and g0 = 6, first calculate their arithmetic mean and geometricmean, thus:

a1 = 12 (24 + 6) = 15

g1 =√24× 6 = 12

and then iterate as follows:

a2 = 12 (15 + 12) = 13.5

g2 =√15× 12 = 13.41640786500 . . .

. . .

The first five iterations give the following values:

As can be seen, the number of digits in agreement (underlined) approximately doubles with each iteration. Thearithmetic–geometricmean of 24 and 6 is the common limit of these two sequences, which is approximately 13.4581714817256154207668131569743992430538388544.[1]

33

34 CHAPTER 7. ARITHMETIC–GEOMETRIC MEAN

7.2 History

The first algorithm based on this sequence pair appeared in the works of Lagrange. Its properties were furtheranalyzed by Gauss.[2]

7.3 Properties

The geometric mean of two positive numbers is never bigger than the arithmetic mean (see inequality of arithmeticand geometric means); as a consequence, (gn) is an increasing sequence, (an) is a decreasing sequence, and gn ≤M(x,y) ≤ an. These are strict inequalities if x ≠ y.M(x, y) is thus a number between the geometric and arithmetic mean of x and y; in particular it is between x and y.If r ≥ 0, then M(rx,ry) = r M(x,y).There is an integral-form expression for M(x,y):

M(x, y) =π

2

/∫ π2

0

dθ√x2 cos2 θ + y2 sin2 θ

=π

4· x+ y

K(

x−yx+y

)where K(k) is the complete elliptic integral of the first kind:

K(k) =

∫ π2

0

dθ√1− k2 sin2(θ)

Indeed, since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute ellipticintegrals via this formula. In engineering, it is used for instance in elliptic filter design.[3]

7.4 Related concepts

The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is called Gauss’s constant, after CarlFriedrich Gauss.

1

M(1,√2)

= G = 0.8346268 . . .

The geometric–harmonicmean can be calculated by an analogousmethod, using sequences of geometric and harmonicmeans. The arithmetic–harmonic mean can be similarly defined, but takes the same value as the geometric mean.The arithmetic–geometric mean can be used to compute logarithms and complete elliptic integrals of the first kind.A modified arithmetic–geometric mean can be used to efficiently compute complete elliptic integrals of the secondkind.[4]

7.5 Proof of existence

From inequality of arithmetic and geometric means we can conclude that:

gn ⩽ an

7.6. PROOF OF THE INTEGRAL-FORM EXPRESSION 35

and thus

gn+1 =√gn · an ⩾ √

gn · gn = gn

that is, the sequence gn is nondecreasing.Furthermore, it is easy to see that it is also bounded above by the larger of x and y (which follows from the fact thatboth arithmetic and geometric means of two numbers both lie between them). Thus by the monotone convergencetheorem the sequence is convergent, so there exists a g such that:

limn→∞

gn = g

However, we can also see that:

an =g2n+1

gn

and so:

limn→∞

an = limn→∞

g2n+1

gn=

g2

g= g

Q.E.D.

7.6 Proof of the integral-form expression

This proof is given by Gauss.[2] Let

I(x, y) =

∫ π/2

0

dθ√x2 cos2 θ + y2 sin2 θ

,

Changing the variable of integration to θ′ , where

sin θ =2x sin θ′

(x+ y) + (x− y) sin2 θ′,

gives

I(x, y) =

∫ π/2

0

dθ′√(12 (x+ y)

)2 cos2 θ′ + (√xy

)2 sin2 θ′= I

(12 (x+ y),

√xy

).

Thus, we have

I(x, y) = I(a1, g1) = I(a2, g2) = · · ·= I

(M(x, y),M(x, y)

)= π/

(2M(x, y)

).

The last equality comes from observing that I(z, z) = π/(2z) .Finally, we obtain the desired result

M(x, y) = π/(2I(x, y)

).

36 CHAPTER 7. ARITHMETIC–GEOMETRIC MEAN

7.7 See also• Generalized mean

• Inequality of arithmetic and geometric means

• Gauss–Legendre algorithm

7.8 External links• Arithmetic-Geometric Mean Calculator

• Proof of convergence rate in PlanetMath

7.9 References• Adlaj, Semjon (September 2012). “An eloquent formula for the perimeter of an ellipse” (PDF). Notices of the

AMS 59 (8): 1094–1099. doi:10.1090/noti879.

• Jonathan Borwein, Peter Borwein, Pi and the AGM. A study in analytic number theory and computationalcomplexity. Reprint of the 1987 original. CanadianMathematical Society Series of Monographs and AdvancedTexts, 4. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. xvi+414 pp. ISBN0-471-31515-X MR 1641658

• Zoltán Daróczy, Zsolt Páles, Gauss-composition of means and the solution of the Matkowski–Suto problem.Publ. Math. Debrecen 61/1-2 (2002), 157–218.

• M.Hazewinkel (2001), “Arithmetic–geometric mean process”, in Hazewinkel, Michiel, Encyclopedia of Math-ematics, Springer, ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Arithmetic–Geometric mean”, MathWorld.

[1] agm(24, 6) at WolframAlpha

[2] David A. Cox (2004). “The Arithmetic-Geometric Mean of Gauss”. In J.L. Berggren, Jonathan M. Borwein, Peter Bor-wein. Pi: A Source Book. Springer. p. 481. ISBN 978-0-387-20571-7. first published in L'Enseignement Mathématique, t.30 (1984), p. 275-330

[3] Hercules G. Dimopoulos (2011). Analog Electronic Filters: Theory, Design and Synthesis. Springer. pp. 147–155. ISBN978-94-007-2189-0.

[4] Adlaj, Semjon (September 2012), “An eloquent formula for the perimeter of an ellipse” (PDF), Notices of the AMS 59 (8):1094–1099, doi:10.1090/noti879, retrieved 2013-12-14

Chapter 8

Askey–Gasper inequality

In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey andGeorge Gasper (1976) and used in the proof of the Bieberbach conjecture.

8.1 Statement

It states that if β ≥ 0, α + β ≥ −2, and −1 ≤ x ≤ 1 then

n∑k=0

P(α,β)k (x)

P(β,α)k (1)

≥ 0

where

P(α,β)k (x)

is a Jacobi polynomial.The case when β = 0 can also be written as

3F2

(−n, n+ α+ 2, 1

2 (α+ 1); 12 (α+ 3), α+ 1; t

)> 0, 0 ≤ t < 1, α > −1.

In this form, with α a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbachconjecture.

8.2 Proof

Ekhad (1993) gave a short proof of this inequality, by combining the identity

(α+ 2)nn!

× 3F2

(−n, n+ α+ 2, 1

2 (α+ 1); 12 (α+ 3), α+ 1; t

)=

=

(12

)j

(α2 + 1

)n−j

(α2 + 3

2

)n−2j

(α+ 1)n−2j

j!(α2 + 3

2

)n−j

(α2 + 1

2

)n−2j

(n− 2j)!× 3F2

(−n+ 2j, n− 2j + α+ 1, 1

2 (α+ 1); 12 (α+ 2), α+ 1; t

)with the Clausen inequality.

37

38 CHAPTER 8. ASKEY–GASPER INEQUALITY

8.3 Generalizations

Gasper & Rahman (2004, 8.9) give some generalizations of the Askey–Gasper inequality to basic hypergeometricseries.

8.4 See also• Turán’s inequalities

8.5 References• Askey, Richard; Gasper, George (1976), “Positive Jacobi polynomial sums. II”, American Journal of Math-

ematics (American Journal of Mathematics, Vol. 98, No. 3) 98 (3): 709–737, doi:10.2307/2373813, ISSN0002-9327, JSTOR 2373813, MR 0430358

• Askey, Richard; Gasper, George (1986), “Inequalities for polynomials”, in Baernstein, Albert; Drasin, David;Duren, Peter; Marden, Albert, The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr.21, Providence, R.I.: American Mathematical Society, pp. 7–32, ISBN 978-0-8218-1521-2, MR 875228

• Ekhad, Shalosh B. (1993), Delest, M.; Jacob, G.; Leroux, P., eds., “A short, elementary, and easy, WZ proof oftheAskey-Gasper inequality that was used by de Branges in his proof of the Bieberbach conjecture”, TheoreticalComputer Science, Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991) 117(1): 199–202, doi:10.1016/0304-3975(93)90313-I, ISSN 0304-3975, MR 1235178

• Gasper, George (2004), Basic hypergeometric series, Encyclopedia ofMathematics and its Applications 96 (2nded.), Cambridge University Press, doi:10.2277/0521833574 (inactive 2015-01-10), ISBN 978-0-521-83357-8, MR 2128719 |first2= missing |last2= in Authors list (help)

Chapter 9

List of special functions and eponyms

This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differentialequations they satisfy, named differential operators of the theory (but not intended to include every mathematicaleponym). Named symmetric functions, and other special polynomials, are included.Contents :

• Top

• 0–9

• A

• B

• C

• D

• E

• F

• G

• H

• I

• J

• K

• L

• M

• N

• O

• P

• Q

• R

• S

• T

39

40 CHAPTER 9. LIST OF SPECIAL FUNCTIONS AND EPONYMS

• U

• V

• W

• X

• Y

• Z

9.1 A

• Niels Abel: Abel polynomials - Abelian function - Abel–Gontscharoff interpolating polynomial

• Sir George Biddell Airy: Airy function

• Waleed Al-Salam (1926–1996): Al-Salam polynomial - Al Salam–Carlitz polynomial - Al Salam–Chiharapolynomial

• C. T. Anger: Anger–Weber function

• Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral

• Paul Émile Appell (1855–1930): Appell hypergeometric series, Appell polynomial, Generalized Appell poly-nomials

• Richard Askey: Askey–Wilson polynomial, Askey–Wilson function (with James A. Wilson)

9.2 B

• Ernest William Barnes: Barnes G-function

• E. T. Bell: Bell polynomials

• Bender–Dunne polynomial

• Jacob Bernoulli: Bernoulli polynomial

• Friedrich Bessel: Bessel function, Bessel–Clifford function

• H. Blasius: Blasius functions

• R. P. Boas, R. C. Buck: Boas–Buck polynomial

• Böhmer integral

• de Bruijn function

• Buchstab function

• Burchnall, Chaundy: Burchnall–Chaundy polynomial

9.3. C 41

9.3 C• Leonard Carlitz: Carlitz polynomial

• Arthur Cayley, Capelli: Cayley–Capelli operator

• Celine’s polynomial• Charlier polynomial

• Pafnuty Chebyshev: Chebyshev polynomials

• Christoffel, Darboux: Christoffel–Darboux relation

• Cyclotomic polynomials

9.4 D• Dawson function

• Charles F. Dunkl: Dunkl operator, Jacobi–Dunkl operator, Dunkl–Cherednik operator

• Dickman–de Bruijn function

9.5 E• Engel: Engel expansion

• Erdélyi Artúr: Erdelyi–Kober operator

• Leonhard Euler: Euler polynomial, Eulerian integral, Euler hypergeometric integral

9.6 F• V. N. Faddeeva: Faddeeva function (also known as the complex error function; see error function)

9.7 G• C. F. Gauss: Gaussian polynomial, Gaussian distribution, etc.

• Leopold Bernhard Gegenbauer: Gegenbauer polynomials

• Gottlieb polynomial• Gould polynomial

• Christoph Gudermann: Gudermannian function

9.8 H• Wolfgang Hahn: Hahn polynomial, (with H. Exton) Hahn–Exton Bessel function

• Philip Hall: Hall polynomial, Hall–Littlewood polynomial

• Hermann Hankel: Hankel function

• Heine: Heine functions

42 CHAPTER 9. LIST OF SPECIAL FUNCTIONS AND EPONYMS

• Charles Hermite: Hermite polynomials

• Karl L. W. M. Heun (1859 – 1929): Heun’s equation

• J. Horn: Horn hypergeometric series

• Adolf Hurwitz: Hurwitz zeta-function

9.9 I

9.10 J

• Henry Jack (1917–1978) Dundee: Jack polynomial

• F. H. Jackson: Jackson derivative Jackson integral

• Carl Gustav Jakob Jacobi: Jacobi polynomial

9.11 K

• Joseph Marie Kampe de Feriet (1893–1982): Kampe de Feriet hypergeometric series

• David Kazhdan, George Lusztig: Kazhdan–Lusztig polynomial

• Lord Kelvin: Kelvin function

• Kibble–Slepian formula

• Kirchhoff: Kirchhoff polynomial

• Tom H. Koornwinder: Koornwinder polynomial

• Kostka polynomial, Kostka–Foulkes polynomial

• Mikhail Kravchuk: Kravchuk polynomial

9.12 L

• Edmond Laguerre: Laguerre polynomials

• Gabriel Lamé: Lamé polynomial

• G. Lauricella Lauricella-Saran: Lauricella hypergeometric series

• Adrien-Marie Legendre: Legendre polynomials

• EugenCornelius Joseph von Lommel (1837–1899), physicist: Lommel polynomial, Lommel function, Lommel–Weber function

9.13 M

• Ian G. Macdonald: Macdonald polynomial, Macdonald–Kostka polynomial, Macdonald spherical function

• Mahler polynomial• Maitland function

• Émile Léonard Mathieu: Mathieu function

9.14. P 43

• F. G. Mehler, student of Dirichlet (Ferdinand): Mehler’s formula, Mehler–Fock formula, Mehler–Heine for-mula, Meier function

• Meijer G-function

• Josef Meixner: Meixner polynomial, Meixner-Pollaczek polynomial

• Mittag-Leffler: Mittag-Leffler polynomials

• Mott polynomial

9.14 P

• Paul Painlevé: Painlevé function, Painlevé transcendents

• Poisson–Charlier polynomial

• Pollaczek polynomial

9.15 R

• Giulio Racah: Racah polynomial

• Jacopo Riccati: Riccati–Bessel function

• Bernhard Riemann: Riemann zeta-function

• Olinde Rodrigues: Rodrigues formula

• Leonard James Rogers: Rogers–Askey–Ismail polynomial, Rogers–Ramanujan identity, Rogers–Szegő poly-nomial

9.16 S

• Schubert polynomial

• Issai Schur: Schur polynomial

• Atle Selberg: Selberg integral

• Sheffer polynomial• Slater’s identities

• Thomas Joannes Stieltjes: Stieltjes polynomial, Stieltjes–Wigert polynomials

• Strömgren function

• Hermann Struve: Struve function

9.17 T

• Francesco Tricomi: Tricomi–Carlitz polynomial

44 CHAPTER 9. LIST OF SPECIAL FUNCTIONS AND EPONYMS

9.18 W• Wall polynomial

• Wangerein: Wangerein functions

• Weber function

• Weierstrass: Weierstrass function

• Louis Weisner: Weisner’s method

• E. T. Whittaker: Whittaker function

• Wilson polynomial

9.19 Z• Frits Zernike: Zernike polynomials

Chapter 10

Special functions

Special functions are particular mathematical functions which have more or less established names and notationsdue to their importance in mathematical analysis, functional analysis, physics, or other applications.There is no general formal definition, but the list of mathematical functions contains functions which are commonlyaccepted as special.

10.1 Tables of special functions

Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore,tables of integrals [1] usually include descriptions of special functions, and tables of special functions [2] includemost important integrals; at least, the integral representation of special functions. Because symmetries of differentialequations are essential to both physics and mathematics, the theory of special functions is closely related to the theoryof Lie groups and Lie algebras, as well as certain topics in mathematical physics.Symbolic computation engines usually recognize the majority of special functions. Not all such systems have efficientalgorithms for the evaluation, especially in the complex plane.

10.1.1 Notations used for special functions

Functions with established international notations are sin, cos, exp, erf, and erfc.Some special functions have several notations:

• The natural logarithm can be written Log, log, logₑ or ln, depending on the context.

• The tangent function may be denoted Tan, tan or tg (especially in Russian and Bulgarian literature).

• Arctangent may be called arctan, atan, arctg, or tan−1.

• Bessel functions may be written

• Jn(x),

• besselj(n, x),• BesselJ[n, x].

Subscripts are often used to indicate arguments, typically integers. In a few cases, the semicolon (;) or even backslash(\) is used as a separator. In this case, the translation to algorithmic languages admits ambiguity and may lead toconfusion.Superscripts may indicate not only exponentiation, but modification of a function. Examples include:

• cos3(x) usually indicates (cos(x))3

45

46 CHAPTER 10. SPECIAL FUNCTIONS

• cos2(x) is typically (cos(x))2 , but never cos(cos(x))

• cos−1(x) usually means arccos(x) , and not (cos(x))−1 ; this one typically causes the most confusion, asit is inconsistent with the others.

10.1.2 Evaluation of special functions

Most special functions are considered as a function of a complex variable. They are analytic; the singularities andcuts are described; the differential and integral representations are known and the expansion to the Taylor series orasymptotic series are available. In addition, sometimes there exist relations with other special functions; a complicatedspecial function can be expressed in terms of simpler functions. Various representations can be used for the evaluation;the simplest way to evaluate a function is to expand it into a Taylor series. However, such representation may convergeslowly if at all. In algorithmic languages, rational approximations are typically used, although they may behave badlyin the case of complex argument(s).

10.2 History of special functions

10.2.1 Classical theory

While trigonometry can be codified — as was clear already to expert mathematicians of the eighteenth century (ifnot before) — the search for a complete and unified theory of special functions has continued since the nineteenthcentury. The high point of special function theory in the period 1800-1900 was the theory of elliptic functions;treatises that were essentially complete, such as that of Tannery and Molk, could be written as handbooks to all thebasic identities of the theory. They were based on techniques from complex analysis.From that time onwards it would be assumed that analytic function theory, which had already unified the trigonometricand exponential functions, was a fundamental tool. The end of the century also saw a very detailed discussion ofspherical harmonics.

10.2.2 Changing and fixed motivations

Of course the wish for a broad theory including as many as possible of the known special functions has its intellectualappeal, but it is worth noting other motivations. For a long time, the special functions were in the particular provinceof applied mathematics; applications to the physical sciences and engineering determined the relative importanceof functions. In the days before the electronic computer, the ultimate compliment to a special function was thecomputation, by hand, of extended tables of its values. This was a capital-intensive process, intended to make thefunction available by look-up, as for the familiar logarithm tables. The aspects of the theory that then mattered mightthen be two:

• for numerical analysis, discovery of infinite series or other analytical expression allowing rapid calculation; and

• reduction of as many functions as possible to the given function.

In contrast, one might say, there are approaches typical of the interests of pure mathematics: asymptotic analysis,analytic continuation and monodromy in the complex plane, and the discovery of symmetry principles and otherstructure behind the façade of endless formulae in rows. There is not a real conflict between these approaches, in fact.

10.2.3 Twentieth century

The twentieth century saw several waves of interest in special function theory. The classic Whittaker and Watson(1902) textbook sought to unify the theory by using complex variables; the G. N. Watson tome A Treatise on theTheory of Bessel Functions pushed the techniques as far as possible for one important type that particularly admittedasymptotics to be studied.The later Bateman Manuscript Project, under the editorship of Arthur Erdélyi, attempted to be encyclopedic, andcame around the time when electronic computation was coming to the fore and tabulation ceased to be the main issue.

10.3. SPECIAL FUNCTIONS IN NUMBER THEORY 47

10.2.4 Contemporary theories

The modern theory of orthogonal polynomials is of a definite but limited scope. Hypergeometric series became anintricate theory, in need of later conceptual arrangement. Lie groups, and in particular their representation theory,explain what a spherical function can be in general; from 1950 onwards substantial parts of classical theory couldbe recast in terms of Lie groups. Further, work on algebraic combinatorics also revived interest in older parts ofthe theory. Conjectures of Ian G. Macdonald helped to open up large and active new fields with the typical specialfunction flavour. Difference equations have begun to take their place besides differential equations as a source forspecial functions.

10.3 Special functions in number theory

In number theory, certain special functions have traditionally been studied, such as particular Dirichlet series andmodular forms. Almost all aspects of special function theory are reflected there, as well as some new ones, such ascame out of the monstrous moonshine theory.

10.4 See also• List of mathematical functions

• List of special functions and eponyms

10.5 References[1] Gradshteyn, I. S.; I. M. Ryzhik (2007). Table of Integrals, Series, and Products. Academic press.

[2] Abramowitz, Milton; Irene A. Stegun (1964). Handbook of Mathematical Functions.

• Andrews, George E.; Askey, Richard; Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematicsand its Applications 71. Cambridge University Press. ISBN 978-0-521-62321-6. MR 1688958.

10.6 External links• Special functions, Special functions calculator.

• Online calculator, Online scientific calculator with over 100 functions (>=32 digits, many complex)

• Special functions at EqWorld: The World of Mathematical Equations.

• Special functions and polynomials’ ' by 't Hooft and Nobbenhuis

• NIST Digital Library of Mathematical Functions, Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert,and Charles W. Clark (Editors) (2010).

• Numerical Methods for Special Functions, by A. Gil, J. Segura, N.M. Temme (2007).

• R. Jagannathan, (P,Q)-Special Functions

• Specialfunctionswiki

48 CHAPTER 10. SPECIAL FUNCTIONS

10.7 Text and image sources, contributors, and licenses

10.7.1 Text• Abramowitz and Stegun Source: https://en.wikipedia.org/wiki/Abramowitz_and_Stegun?oldid=671365541 Contributors: Arpingstone,

Stevenj, Charles Matthews, Jitse Niesen, Taxman, MathMartin, Robinh, Giftlite, Langec, MarkSweep, DragonflySixtyseven, Almit39,Neutrality, MakeRocketGoNow, Rich Farmbrough, ArnoldReinhold, Oleg Alexandrov, Linas, NeoUrfahraner, Rjwilmsi, R.e.b., Jdfor-resterBot, Kerry Raymond, Zvika, JJL, SmackBot, RDBury, InverseHypercube, Ephraim33, Derek farn, John, Tkynerd, Catslash, Salih,Agricola44, BotMultichill, Soryukb, Rhubbarb, Qwfp, MystBot, Addbot, Tassedethe, Lightbot, Yobot, Specfunfan, CobraBot, Rjwilm-siBot, ZéroBot, Sdocan, OccultZone and Anonymous: 12

• Absolute value Source: https://en.wikipedia.org/wiki/Absolute_value?oldid=670579867 Contributors: AxelBoldt, Bryan Derksen, Zun-dark, Tarquin, Andre Engels, Edemaine, Heron, Edward, Patrick, JohnOwens, Ixfd64, Dcljr, TakuyaMurata, Cgs, Rossami, Andres, Ever-cat, Pizza Puzzle, Schneelocke, Hashar, Revolver, Dcoetzee, LennyWikipedia~enwiki, Dysprosia, JitseNiesen, Furrykef, Itai, Moriel~enwiki,Rogper~enwiki, Chuunen Baka, Robbot, Rfc1394, Henrygb, Robinh, Mfc, Giftlite, Gene Ward Smith, BenFrantzDale, Ævar ArnfjörðBjarmason, Lethe, Guanaco, Yekrats, Jason Quinn, Eequor, Macrakis, OTB, Andycjp, DocSigma, LucasVB, Antandrus, Histrion, Hk-pawn~enwiki, Ukexpat, Karl Dickman, Qef, Mernen, Rich Farmbrough, Guanabot, Inkypaws, MeltBanana, Ivan Bajlo, Paul August,Syp, Rgdboer, Art LaPella, EmilJ, Spoon!, .:Ajvol:., Nk, Alansohn, ABCD, Caesura, Sleigh, Drbreznjev, InBalance, Oleg Alexandrov,Blumpkin, Rocastelo, Matijap, MFH, The wub, VKokielov, RobertG, Glenn L, Chobot, DVdm, YurikBot, Laurentius, Dmharvey, KSmrq,CambridgeBayWeather, NawlinWiki, Gwaihir, Vanished user 1029384756, Jimmyre, Bota47, Jessemerriman, Mike92591, Igiffin, Zzu-uzz, ArielGold, Gesslein, That Guy, From That Show!, SmackBot, RDBury, Incnis Mrsi, InverseHypercube, Pgk, BiT, Diegotorque-mada, Aksi great, Betacommand, ERcheck, Jweimar, Silly rabbit, Adpete, DHN-bot~enwiki, Emorgasm, Wen D House, Cybercobra,Nakon, Metebelis, LN2, Breno, Minna Sora no Shita, Jim.belk, A.Z., IronGargoyle, Ckatz, KJS77, Iridescent, Madmath789, Splitpea-soup, The editor1, AlainD, CRGreathouse, CmdrObot, FilipeS, Doctormatt, Karimarie, Gogo Dodo, He Who Is, Khattab01~enwiki,FastLizard4, Xantharius, Epbr123, LeeG, Headbomb, Marek69, Futurebird, Mentifisto, AntiVandalBot, Edokter, GiM, Hannes Eder,Dylan Lake, Rower2000, Acroterion, Magioladitis, Bongwarrior, VoABot II, Brunoman1990, Cic, David Eppstein, Gwern, FisherQueen,MartinBot, Ortensia, Highegg, Bracodbk, Avatar09, Chrishy man, J.delanoy, Defilerc, Spowage, Jacksonwalters, JonMcLoone, Policron,Aephoenix, Mviergujerghs89fhsdifds, Jamesofur, Mathsexpressions, Useight, JohnDoe0007, Idioma-bot, VolkovBot, Aesopos, PhilipTrueman, TXiKiBoT, Ann Stouter, Anonymous Dissident, JhsBot, Cremepuff222, Nibios, Dmcq, AlleborgoBot, SieBot, Yintan, Jab-baTheBot, JackSchmidt, OKBot, M2Ys4U, Escape Orbit, Dlrohrer2003, Atif.t2, ClueBot, Brewcrewer, Coriakin~enwiki, Excirial, Polly,Calor, 7, BlueDevil, Kiensvay, Spitfire, ZooFari, Thatguyflint, Fgnievinski, Ronhjones, WMdeMuynck, With goodness in mind, The worlddeserves the truth, AndersBot, SpBot, Ginosbot, Tide rolls, Zorrobot, LuK3, Legobot, Luckas-bot, Yobot, THENWHOWAS PHONE?,AnomieBOT, Jim1138, Piano non troppo, Yachtsman1, JohnnyB256, Jxramos, ArthurBot, DannyAsher, Hanberke, Isheden, Inferno,Lord of Penguins, Jhbdel, Amaury, VoItorb, FrescoBot, Majopius, Hamtechperson, SpaceFlight89, Yaddie, White Shadows, TobeBot,General Helper, Ammodramus, Aniten21, RjwilmsiBot, Q6913, Midhart90, EcneicsFlogCitanaf, Salvio giuliano, EmausBot, Frankjohnli,Tommy2010, Wikipelli, Slawekb, Robirahman, AvicBot, John Cline, Traxs7, C0rtesf, D.Lazard, Wayne Slam, Mcmatter, Tolly4bolly,Cit helper, TyA, Theonefoster, Vladimirdx, Chewings72, Nickminaj15, ResearchRave, ClueBot NG, KlappCK, Priyankastar, Mesoderm,O.Koslowski, Joel B. Lewis, Savantas83, Helpful Pixie Bot, Drummerboy5324, Mrjohncummings, Achowat, Davidfreesefan23, Rock-hand, Tutelary, Henri.vanliempt, ChrisGualtieri, Travelpleb, Webclient101, Wiki2487, Conner11108, Conner111088, Jareknh, DrakeLehto, Evagavilan, Whikie, Wigglewigglewiggle3, SoSivr, Luisrafael1221, Milesoc and Anonymous: 269

• Ackermann function Source: https://en.wikipedia.org/wiki/Ackermann_function?oldid=670242294 Contributors: AxelBoldt, DerekRoss, Mav, Matusz, Edemaine, Mjb, Patrick, Michael Hardy, SGBailey, Ixfd64, Chinju, Cyde, Paul A, Rochus, Cyp, Ootachi, Susurrus,Schneelocke, Alexwatson, Timwi, Dcoetzee, Ww, Dysprosia, Doradus, VeryVerily, Populus, Kwantus, Raul654, Pakaran, Jerzy, Fredrik,Romanm, Chris Roy, MathMartin, Ashley Y, Thesilverbail, DHN, Bkell, Ruakh, Tobias Bergemann, David Gerard, Giftlite, Gene WardSmith, Adam Sampson, Lupin, Herbee, Dissident, Anville, Gro-Tsen, Elias, Sundar, Prosfilaes, Pne, Bobblewik, Xinit, MarkDettinger,Pmanderson, Grunt, Mormegil, Guanabot, Leibniz, Ardonik, Ponder, Night Gyr, Ben Standeven, Pt, Susvolans, Larry V, ObradovicGoran, Alansohn, TILL, Sligocki, Burn, Wdfarmer, Sveno, Vedantm, Earpol, RJFJR, Pgimeno~enwiki, Drbreznjev, Oleg Alexandrov,Gmaxwell, Daniel Geisler, Bo198214~enwiki, StradivariusTV, Ruud Koot, GregorB, Graham87, Rjwilmsi, MarSch, Eyu100, Salixalba, Mike Segal, R.e.b., Bubba73, FlaBot, Margosbot~enwiki, NekoDaemon, Algebraist, Roboto de Ajvol, YurikBot, Hairy Dude,Freiberg, FrenchIsAwesome, Iamfscked, Gaius Cornelius, Think Fast, Trovatore, R.e.s., Joelr31, Hv, SandBoxer, Ott2, CharlesHBen-nett, Gesslein, PhS, GrinBot~enwiki, KnightRider~enwiki, SmackBot, Maksim-e~enwiki, Incnis Mrsi, Rmeht, Betacommand, Ppntori,Bluebot, Irving Anellis, Acipsen, DHN-bot~enwiki, Emurphy42, Kaimiddleton, Short Circuit, Cybercobra, Nakon, Shadow1, DMacks,TenPoundHammer, Lambiam, Derek farn, Harryboyles, Draco18s, Rsimmonds01, Ryszard Szopa~enwiki, Wrstark, Physis, Loadmas-ter, Rainwarrior, Pocketfox, Alkanen, Stephen B Streater, Grimboy, Gabn1, R~enwiki, Domitori, JRSpriggs, CRGreathouse, CBM,Pierre de Lyon, Lentower, Cydebot, Yrodro, MotherFunctor, Quibik, Kristofer0, Eowend, JMatthews, Dfrg.msc, Escarbot, Kylemcinnes,Kamocat, Yomangani, Braindrain0000, Christopher.lck, Myanw, Turgidson, Cameron.walsh, .anacondabot, Magioladitis, David Epp-stein, JaGa, Glrx, F0rked, It Is Me Here, Plasticup, Policron, User77764, PGSONIC, Af648, Mdmkolbe, TXiKiBoT, Yugsdrawkcabeht,Anonymous Dissident, The Wilschon, Farever, MadnessRed, Jesin, AJRobbins, AlleborgoBot, SieBot, Taemyr, Eshui2007, Ubermam-mal, Chx1975, JackSchmidt, OKBot, Anchor Link Bot, Ehsansherkat, Phoetus, ClueBot, The Thing That Should Not Be, Nguyenhh,Adrianwn, Mild Bill Hiccup, Tuchomator, Absolt, XLinkBot, Marc van Leeuwen, WillOakland, Brentsmith101, Fieshp, Addbot, DOIbot, Pietrow, ,ماني Yobot, AnomieBOT, Blue Prawn~enwiki, Materialscientist, Xymostech, Aleph Infinity, JmCor, Xqbot, Nmesisgeek,WebCiteBOT, Fredericksgary, MathHisSci, Incantar, Citation bot 1, Double sharp, Trappist the monk, Melara..., Updatehelper, Chaoticdemon, Derekleungtszhei, Didym, Δ, ClueBot NG, JOHNDOE, Dexbot, Zziccardi, LegionMammal978, Stamptrader, Monkbot, Wjxband Anonymous: 181

• Airy function Source: https://en.wikipedia.org/wiki/Airy_function?oldid=665272879 Contributors: Gareth Owen, Michael Hardy, Cyp,Schneelocke, Emperorbma, Charles Matthews, Dysprosia, Jitse Niesen, Tobias Bergemann, Giftlite, Sj, Alberto da Calvairate~enwiki,Creidieki, Eric Kvaalen, PAR, Oleg Alexandrov, Linas, Jftsang, BD2412, R.e.b., Alejo2083, YurikBot, KSmrq, Gaius Cornelius, ArthurRubin, Katieh5584, InverseHypercube, Diegotorquemada, Njerseyguy, Papa November, Dchristle, Thijs!bot, Headbomb, Hannes Eder,WordSurd, Stevvers, Krishnachandranvn, Haseldon, Policron, VolkovBot, Free0willy, Sunrise, DragonBot, Addbot, Josuedavila, PV=nRT,Luckas-bot, Yobot, MichalKotowski, Geek1337~enwiki, Xqbot, Bdmy, Olaf3142, FrescoBot, Ichbin-dcw, Citation bot 1, SUL, Tcnuk,Nerissa-Marie, Netheril96, ZéroBot, Electron0511, El Roih, ServiceAT, Jovianic, Fuse809, Purdygb, Mediran, OpticalEngineer, Fazhbr,Jamali.ehsan1982, Davidsalmon1, Njsbling, Pjgaudre and Anonymous: 27

10.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 49

• Algebraic function Source: https://en.wikipedia.org/wiki/Algebraic_function?oldid=673260293 Contributors: Michael Hardy, CharlesMatthews, Haukurth, Giftlite, GeneWard Smith, Wmahan, Almit39, GrAfFiT, Photonique, Ksnow, Waldir, MarSch, Margosbot~enwiki,YurikBot, SmackBot, InverseHypercube, Bluebot, Bethling, Silly rabbit, Tsca.bot, BenWhitey, Krashlandon, Cronholm144, Mets501,Valoem, CBM, Dugwiki, AntiVandalBot, VictorAnyakin, Dekimasu, Johnbibby, MetsBot, Pomte, AlleborgoBot, Kai Su?, EmxBot,SieBot, VVVBot, JackSchmidt, ClueBot, PipepBot, JP.Martin-Flatin, DumZiBoT,Mattsem, Addbot, PV=nRT, Luckas-bot, AnomieBOT,Ciphers, Wisems, Isheden, GrouchoBot, Sławomir Biała, EmausBot, WikitanvirBot, D.Lazard, ClueBot NG, The1337gamer, JochenBurghardt, Snowright, Elkevn and Anonymous: 46

• Anger function Source: https://en.wikipedia.org/wiki/Anger_function?oldid=601733807 Contributors: Michael Hardy, Giftlite, R.e.b.,Myasuda, Headbomb, Sikory, Jiuguang Wang, Crowsnest, Addbot, LaaknorBot, Ildeguz, Suslindisambiguator, ChuispastonBot, El Roih,KLBot2, Lamarit and Makecat-bot

• Arithmetic–geometric mean Source: https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean?oldid=664174974 Con-tributors: Damian Yerrick, AxelBoldt, FvdP, Michael Hardy, Ejrh, Poor Yorick, Charles Matthews, Timwi, Dcoetzee, Fredrik, Gan-dalf61, Henrygb, Giftlite, Kenny TM~~enwiki, Rich Farmbrough, Guanabot, Jason Davies, Sligocki, Oleg Alexandrov, Linas, Shree-vatsa, Rjwilmsi, Harami~enwiki, Nneonneo, Scythe33, Chobot, YurikBot, KSmrq, Gwaihir, Mikeblas, SamuelRiv, GraemeL, Smack-Bot, InverseHypercube, MalafayaBot, Foxjwill, Berland, Meni Rosenfeld, Lambiam, Dicklyon, A. Pichler, Headbomb, Storkk, Asmeurer,A3nm, Gwern, SimpsonDG, Yonidebot, Geometry guy, Arcfrk, Cffk, JackSchmidt, DragonBot, Crowsnest, Kiensvay, Ozob, PV=nRT,Legobot, Luckas-bot, Yobot, TaBOT-zerem, AnomieBOT, Skogssvinet, A. di M., Yaddie, Trappist the monk, Duoduoduo, Tijfo098,Newyorkadam, BG19bot, ChrisGualtieri, Shlav, EefeG0hi, Nigel D Robinson, Everymorning, Ccmcc2012, Bub250, Rafiadler, Mar-tin0shymek and Anonymous: 40

• Askey–Gasper inequality Source: https://en.wikipedia.org/wiki/Askey%E2%80%93Gasper_inequality?oldid=641914571Contributors:Michael Hardy, Jason Quinn, R.e.b., Sodin, A. Pichler, Headbomb, JAnDbot, SieBot, T-rithy, Addbot, Yobot, Citation bot, ArthurBot,Specfunfan, Citation bot 1 and Anonymous: 1

• List of special functions and eponyms Source: https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms?oldid=647297721Contributors: Michael Hardy, Charles Matthews, Topbanana, Gandalf61, Cutler, Giftlite, Linas, Nneonneo, R.e.b., Mathbot, Kmorozov,Dstrozzi, Welsh, Jushi, Nick Levine, Syrcatbot, CRGreathouse, Myasuda, Thijs!bot, Tryptographer, Magioladitis, Vanish2, SJP, Cuzkatz-imhut, AJRobbins, Niceguyedc, Ultra.Power, Addbot, Math1353, Tassedethe, Yobot, SporkBot, K9re11 and Anonymous: 7

• Special functions Source: https://en.wikipedia.org/wiki/Special_functions?oldid=668378136 Contributors: Gianfranco, Michael Hardy,Graue, Charles Matthews, Dysprosia, Fredrik, Romanm, Gandalf61, MathMartin, Filemon, Giftlite, Macrakis, DemonThing, SURIV,Almit39, Abdull, Shahab, Oleg Alexandrov, Waabu, Tbsmith, Linas, Madmardigan53, IIBewegung, Qwertyus, R.e.b., Chobot, Roboto deAjvol, YurikBot, SmackBot, Lambiam, Hvn0413, Owlbuster, Bhaskara~enwiki, Domitori, Khromegnome,MaxEnt, Thijs!bot, HowiAuck-land, Coppertwig, Jamelan, AJRobbins, SieBot, HairyFotr, DumZiBoT, Soulzone, Addbot, Dr. Universe, Favonian, Ehrenkater, Light-bot, Luckas-bot, Ptbotgourou, Rubinbot, Isheden, RibotBOT, FrescoBot, D'ohBot, Citation bot 1, Trappist the monk, Xnn, Asymptoti-cus, Suslindisambiguator, Ocean Shores, Sameenahmedkhan, Mark viking, Yardimsever, Monkbot, Degenerate prodigy, KasparBot andAnonymous: 39

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50 CHAPTER 10. SPECIAL FUNCTIONS

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