8132320722.HandbookMathFunctions

Table of Contents

Chapter 1 - Introduction to Function

Chapter 2 - Inverse Function

Chapter 3 - Special Functions & Implicit and Explicit Functions

Chapter 4 - Function Composition

Chapter 5 - Continuous Function

Chapter 6 - Additive Function Chapter 7 - Algebraic Function Chapter 8 - Analytic Function Chapter 9 - Completely Multiplicative Function and Concave Function

Chapter 10 - Convex Function

Chapter 11 - Differentiable Function

Chapter 12 - Elementary Function and Entire Function

Chapter 13 - Even and Odd Functions

Chapter 14 - Harmonic Function

Chapter 15 - Holomorphic Function

Chapter 16 - Homogeneous Function

Chapter 17 - Indicator Function

Chapter 18 - Injective Function

Chapter 19 - Measurable Function

Chapter 1

Introduction to Function

Graph of example function,

Both the domain and the range in the picture are the set of real numbers between -1 and 1.5.

The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain of the function. An example of a function with the real numbers as both its domain and codomain is the function f(x) =

2x, which assigns to every real number the real number with twice its value. In this case, it is written that f(5) = 10.

In addition to elementary functions on numbers, functions include maps between algebraic structures like groups and maps between geometric objects like manifolds. In the abstract set-theoretic approach, a function is a relation between the domain and the codomain that associates each element in the domain with exactly one element in the codomain. An example of a function with domain {A,B,C} and codomain {1,2,3} associates A with 1, B with 2, and C with 3.

There are many ways to describe or represent functions: by a formula, by an algorithm that computes it, by a plot or a graph. A table of values is a common way to specify a function in statistics, physics, chemistry, and other sciences. A function may also be described through its relationship to other functions, for example, as the inverse function or a solution of a differential equation. There are uncountably many different functions from the set of natural numbers to itself, most of which cannot be expressed with a formula or an algorithm.

In a setting where they have numerical outputs, functions may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually required to be closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is the composition of functions.

Overview Because functions are so widely used, many traditions have grown up around their use. The symbol for the input to a function is often called the independent variable or argument and is often represented by the letter x or, if the input is a particular time, by the letter t. The symbol for the output is called the dependent variable or value and is often represented by the letter y. The function itself is most often called f, and thus the notation y = f(x) indicates that a function named f has an input named x and an output named y.

A function takes an input, x, and returns an output (x). One metaphor describes the function as a "machine" or "black box" that converts the input into the output.

The set of all permitted inputs to a given function is called the domain of the function. The set of all resulting outputs is called the image or range of the function. The image is often a subset of some larger set, called the codomain of a function. Thus, for example, the function f(x) = x2 could take as its domain the set of all real numbers, as its image the set of all non-negative real numbers, and as its codomain the set of all real numbers. In that case, we would describe f as a real-valued function of a real variable. Sometimes, especially in computer science, the term "range" refers to the codomain rather than the image, so care needs to be taken when using the word.

It is usual practice in mathematics to introduce functions with temporary names like . For example, (x) = 2x+1, implies (3) = 7; when a name for the function is not needed, the form y = 2x+1 may be used. If a function is often used, it may be given a more permanent name as, for example,

Functions need not act on numbers: the domain and codomain of a function may be arbitrary sets. One example of a function that acts on non-numeric inputs takes English words as inputs and returns the first letter of the input word as output. Furthermore, functions need not be described by any expression, rule or algorithm: indeed, in some cases it may be impossible to define such a rule. For example, the association between inputs and outputs in a choice function often lacks any fixed rule, although each input element is still associated to one and only one output.

A function of two or more variables is considered in formal mathematics as having a domain consisting of ordered pairs or tuples of the argument values. For example Sum(x,y) = x+y operating on integers is the function Sum with a domain consisting of pairs of integers. Sum then has a domain consisting of elements like (3,4), a codomain of integers, and an association between the two that can be described by a set of ordered pairs like ((3,4), 7). Evaluating Sum(3,4) then gives the value 7 associated with the pair (3,4).

A family of objects indexed by a set is equivalent to a function. For example, the sequence 1, 1/2, 1/3, ..., 1/n, ... can be written as the ordered sequence where n is a natural number, or as a function f(n) = 1/n from the set of natural numbers into the set of rational numbers.

Dually, a surjective function partitions its domain into disjoint sets indexed by the codomain. This partition is known as the kernel of the function, and the parts are called the fibers or level sets of the function at each element of the codomain. (A non-surjective function divides its domain into disjoint and possibly-empty subsets).

Definition One precise definition of a function is that it consists of an ordered triple of sets, which may be written as (X, Y, F). X is the domain of the function, Y is the codomain, and F is a

set of ordered pairs. In each of these ordered pairs (a, b), the first element a is from the domain, the second element b is from the codomain, and every element in the domain is the first element in one and only one ordered pair. The set of all b is known as the image of the function. Some authors use the term "range" to mean the image, others to mean the codomain.

The notation :XY indicates that is a function with domain X and codomain Y.

In most practical situations, the domain and codomain are understood from context, and only the relationship between the input and output is given. Thus

is usually written as

The graph of a function is its set of ordered pairs. Such a set can be plotted on a pair of coordinate axes; for example, (3, 9) is the point of intersection of the lines x = 3 and y = 9.

A function is a special case of a more general mathematical concept, the relation, for which the restriction that each element of the domain appear as the first element in one and only one ordered pair is removed (or, in other words, the restriction that each input be associated to exactly one output). A relation is "single-valued" or "functional" when for each element of the domain set, the graph contains at most one ordered pair (and possibly none) with it as a first element. A relation is called "left-total" or simply "total" when for each element of the domain, the graph contains at least one ordered pair with it as a first element (and possibly more than one). A relation that is both left-total and single-valued is a function.

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. For example, the function f such that f(x) = 1/x does not define a value for x = 0, and so is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph. In other parts of mathematics, non-single-valued relations are similarly conflated with functions: these are called multivalued functions, with the corresponding term single-valued function for ordinary functions.

Some authors (especially in set theory) define a function as simply its graph f, with the restriction that the graph should not contain two distinct ordered pairs with the same first element. Indeed, given such a graph, one can construct a suitable triple by taking the set of all first elements as the domain and the set of all second elements as the codomain: this automatically causes the function to be total and surjective . However, most authors in

advanced mathematics outside of set theory prefer the greater power of expression afforded by defining a function as an ordered triple of sets.

Many operations in set theorysuch as the power sethave the class of all sets as their domain, therefore, although they are informally described as functions, they do not fit the set-theoretical definition above outlined.

Vocabulary A specific input in a function is called an argument of the function. For each argument value x, the corresponding unique y in the codomain is called the function value at x, output of for an argument x, or the image of x under . The image of x may be written as (x) or as y.

The graph of a function is the set of all ordered pairs (x, (x)), for all x in the domain X. If X and Y are subsets of R, the real numbers, then this definition coincides with the familiar sense of "graph" as a picture or plot of the function, with the ordered pairs being the Cartesian coordinates of points.

A function can also be called a map or a mapping. Some authors, however, use the terms "function" and "map" to refer to different types of functions. Other specific types of functions include functionals and operators.

Notation

Formal description of a function typically involves the function's name, its domain, its codomain, and a rule of correspondence. Thus we frequently see a two-part notation, an example being

where the first part is read:

" is a function from N to R" (one often writes informally "Let : X Y" to mean "Let be a function from X to Y"), or

" is a function on N into R", or " is an R-valued function of an N-valued variable",

and the second part is read:

maps to

Here the function named "" has the natural numbers as domain, the real numbers as codomain, and maps n to itself divided by . Less formally, this long form might be abbreviated

where f(n) is read as "f as function of n" or "f of n". There is some loss of information: we no longer are explicitly given the domain N and codomain R.

It is common to omit the parentheses around the argument when there is little chance of confusion, thus: sin x; this is known as prefix notation. Writing the function after its argument, as in x , is known as postfix notation; for example, the factorial function is customarily written n!, even though its generalization, the gamma function, is written (n). Parentheses are still used to resolve ambiguities and denote precedence, though in some formal settings the consistent use of either prefix or postfix notation eliminates the need for any parentheses.

Functions with multiple inputs and outputs

The concept of function can be extended to an object that takes a combination of two (or more) argument values to a single result. This intuitive concept is formalized by a function whose domain is the Cartesian product of two or more sets.

For example, consider the function that associates two integers to their product: (x, y) = xy. This function can be defined formally as having domain ZZ, the set of all integer pairs; codomain Z; and, for graph, the set of all pairs ((x,y), xy). Note that the first component of any such pair is itself a pair (of integers), while the second component is a single integer.

The function value of the pair (x,y) is ((x,y)). However, it is customary to drop one set of parentheses and consider (x,y) a function of two variables, x and y. Functions of two variables may be plotted on the three-dimensional Cartesian as ordered triples of the form (x,y,f(x,y)).

The concept can still further be extended by considering a function that also produces output that is expressed as several variables. For example, consider the function swap(x, y) = (y, x) with domain RR and codomain RR as well. The pair (y, x) is a single value in the codomain seen as a Cartesian product.

Currying

An alternative approach to handling functions with multiple arguments is to transform them into a chain of functions that each takes a single argument. For instance, one can interpret Add(3,5) to mean "first produce a function that adds 3 to its argument, and then

apply the 'Add 3' function to 5". This transformation is called currying: Add 3 is curry(Add) applied to 3. There is a bijection between the function spaces CAB and (CB)A.

When working with curried functions it is customary to use prefix notation with function application considered left-associative, since juxtaposition of multiple argumentsas in ( x y)naturally maps to evaluation of a curried function.

Binary operations

The familiar binary operations of arithmetic, addition and multiplication, can be viewed as functions from RR to R. This view is generalized in abstract algebra, where n-ary functions are used to model the operations of arbitrary algebraic structures. For example, an abstract group is defined as a set X and a function from XX to X that satisfies certain properties.

Traditionally, addition and multiplication are written in the infix notation: x+y and xy instead of +(x, y) and (x, y).

Injective and surjective functions

Three important kinds of function are the injections (or one-to-one functions), which have the property that if (a) = (b) then a must equal b; the surjections (or onto functions), which have the property that for every y in the codomain there is an x in the domain such that (x) = y; and the bijections, which are both one-to-one and onto. This nomenclature was introduced by the Bourbaki group.

When the definition of a function by its graph only is used, since the codomain is not defined, the "surjection" must be accompanied with a statement about the set the function maps onto. For example, we might say maps onto the set of all real numbers.

Function composition

A composite function g(f(x)) can be visualized as the combination of two "machines". The first takes input x and outputs f(x). The second takes f(x) and outputs g(f(x)).

The function composition of two or more functions takes the output of one or more functions as the input of others. The functions : X Y and g: Y Z can be composed by first applying to an argument x to obtain y = (x) and then applying g to y to obtain z = g(y). The composite function formed in this way from general and g may be written

This notation follows the form such that

The function on the right acts first and the function on the left acts second, reversing English reading order. We remember the order by reading the notation as "g of ". The order is important, because rarely do we get the same result both ways. For example, suppose (x) = x2 and g(x) = x+1. Then g((x)) = x2+1, while (g(x)) = (x+1)2, which is x2+2x+1, a different function.

In a similar way, the function given above by the formula y = 5x20x3+16x5 can be obtained by composing several functions, namely the addition, negation, and multiplication of real numbers.

An alternative to the colon notation, convenient when functions are being composed, writes the function name above the arrow. For example, if is followed by g, where g produces the complex number eix, we may write

A more elaborate form of this is the commutative diagram.

Identity function

The unique function over a set X that maps each element to itself is called the identity function for X, and typically denoted by idX. Each set has its own identity function, so the subscript cannot be omitted unless the set can be inferred from context. Under composition, an identity function is "neutral": if is any function from X to Y, then

Restrictions and extensions

Informally, a restriction of a function is the result of trimming its domain.

More precisely, if is a function from a X to Y, and S is any subset of X, the restriction of to S is the function |S from S to Y such that |S(s) = (s) for all s in S.

If g is a restriction of , then it is said that is an extension of g.

The overriding of f: X Y by g: W Y (also called overriding union) is an extension of g denoted as (f g): (X W) Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. If f|X W and g|X W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

Image of a set

The concept of the image can be extended from the image of a point to the image of a set. If A is any subset of the domain, then (A) is the subset of im consisting of all images of elements of A. We say the (A) is the image of A under f.

Use of (A) to denote the image of a subset AX is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is [A] for the set { (x): x A }; some authors write `x instead of (x), and ``A instead of [A].

Notice that the image of is the image (X) of its domain, and that the image of is a subset of its codomain.

Inverse image

The inverse image (or preimage, or more precisely, complete inverse image) of a subset B of the codomain Y under a function is the subset of the domain X defined by

So, for example, the preimage of {4, 9} under the squaring function is the set {3,2,2,3}.

In general, the preimage of a singleton set (a set with exactly one element) may contain any number of elements. For example, if (x) = 7, then the preimage of {5} is the empty set but the preimage of {7} is the entire domain. Thus the preimage of an element in the codomain is a subset of the domain. The usual convention about the preimage of an element is that 1(b) means 1({b}), i.e

In the same way as for the image, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write 1[B] and 1[b] for the preimage of a set and a singleton.

The preimage of a singleton set is sometimes called a fiber. The term kernel can refer to a number of related concepts.

Specifying a function A function can be defined by any mathematical condition relating each argument to the corresponding output value. If the domain is finite, a function may be defined by simply tabulating all the arguments x and their corresponding function values (x). More commonly, a function is defined by a formula, or (more generally) an algorithm a recipe that tells how to compute the value of (x) given any x in the domain.

There are many other ways of defining functions. Examples include piecewise definitions, induction or recursion, algebraic or analytic closure, limits, analytic continuation, infinite series, and as solutions to integral and differential equations. The

lambda calculus provides a powerful and flexible syntax for defining and combining functions of several variables.

Computability

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

Function spaces The set of all functions from a set X to a set Y is denoted by X Y, by [X Y], or by YX.

The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X Y is |YX| = |Y||X|. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. Other examples are the multiplication sign XY used for the Cartesian product, where |XY| = |X||Y|; the factorial sign X!, used for the set of permutations where

|X!| = |X|!; and the binomial coefficient sign , used for the set of n-element subsets

where

If : X Y, it may reasonably be concluded that [X Y].

Pointwise operations

If : X R and g: X R are functions with a common domain of X and common codomain of a ring R, then the sum function + g: X R and the product function g: X R can be defined as follows:

for all x in X.

This turns the set of all such functions into a ring. The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions. This is an example of climbing up in abstraction, to functions of more complex types.

By taking some other algebraic structure A in the place of R, we can turn the set of all functions from X to A into an algebraic structure of the same type in an analogous way.

Other properties There are many other special classes of functions that are important to particular branches of mathematics, or particular applications. Here is a partial list:

bijection, injection and surjection, or singularly: o injective, o surjective, and o bijective function

continuous differentiable, integrable linear, polynomial, rational algebraic, transcendental trigonometric fractal odd or even convex, monotonic, unimodal holomorphic, meromorphic, entire vector-valued computable

History

Functions prior to Leibniz

Historically, some mathematicians can be regarded as having foreseen and come close to a modern formulation of the concept of function. Among them is Oresme (1323-1382) . . . In his theory, some general ideas about independent and dependent variable quantities seem to be present.

Ponte further notes that "The emergence of a notion of function as an individualized mathematical entity can be traced to the beginnings of infinitesimal calculus".

The notion of "function" in analysis

As a mathematical term, "function" was coined by Gottfried Leibniz, in a 1673 letter, to describe a quantity related to a curve, such as a curve's slope at a specific point. The functions Leibniz considered are today called differentiable functions. For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input. Such functions are the basis of calculus.

Johann Bernoulli "by 1718, had come to regard a function as any expression made up of a variable and some constants", and Leonhard Euler during the mid-18th century used the word to describe an expression or formula involving variables and constants e.g., x2+3x+2.

Alexis Claude Clairaut (in approximately 1734) and Euler introduced the familiar notation " f(x) ".

At first, the idea of a function was rather limited. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. By broadening the definition of functions, mathematicians were able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are, in a precise sense, more common than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion.

During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's.

Dirichlet and Lobachevsky are traditionally credited with independently giving the modern "formal" definition of a function as a relation in which every first element has a unique second element. Eves asserts that "the student of mathematics usually meets the Dirichlet definition of function in his introductory course in calculus, but Dirichlet's claim to this formalization is disputed by Imre Lakatos:

There is no such definition in Dirichlet's works at all. But there is ample evidence that he had no idea of this concept. In his [1837], for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the function has two values: ... (Proofs and Refutations, 151, Cambridge University Press 1976.)

In the context of "the Differential Calculus" George Boole defined (circa 1849) the notion of a function as follows:

"That quantity whose variation is uniform . . . is called the independent variable. That quantity whose variation is referred to the variation of the former is said to be a function of it. The Differential calculus enables us in every case to pass from the function to the limit. This it does by a certain Operation. But in the very Idea of an Operation is . . . the idea of an inverse operation. To effect that inverse operation in the present instance is the business of the Int[egral] Calculus."

The logician's "function" prior to 1850

Logicians of this time were primarily involved with analyzing syllogisms (the 2000 year-old Aristotelian forms and otherwise), or as Augustus De Morgan (1847) stated it: "the examination of that part of reasoning which depends upon the manner in which inferences are formed, and the investigation of general maxims and rules for constructing arguments". At this time the notion of (logical) "function" is not explicit, but at least in the work of De Morgan and George Boole it is implied: we see abstraction of the argument forms, the introduction of variables, the introduction of a symbolic algebra with respect to these variables, and some of the notions of set theory.

De Morgan's 1847 "FORMAL LOGIC OR, The Calculus of Inference, Necessary and Probable" observes that "[a] logical truth depends upon the structure of the statement, and not upon the particular matters spoken of"; he wastes no time (preface page i) abstracting: "In the form of the proposition, the copula is made as absract as the terms". He immediately (p. 1) casts what he calls "the proposition" (present-day propositional function or relation) into a form such as "X is Y", where the symbols X, "is", and Y represent, respectively, the subject, copula, and predicate. While the word "function" does not appear, the notion of "abstraction" is there, "variables" are there, the notion of inclusion in his symbolism all of the is in the (p. 9) is there, and lastly a new symbolism for logical analysis of the notion of "relation" (he uses the word with respect to this example " X)Y " (p. 75)) is there:

" A1 X)Y To take an X it is necessary to take a Y" [or To be an X it is necessary to be a Y] " A1 Y)X To take an Y it is sufficient to take a X" [or To be a Y it is sufficient to be an X], etc.

In his 1848 The Nature of Logic Boole asserts that "logic . . . is in a more especial sense the science of reasoning by signs", and he briefly discusses the notions of "belonging to" and "class": "An individual may possess a great variety of attributes and thus belonging to a great variety of different classes" . Like De Morgan he uses the notion of "variable" drawn from analysis; he gives an example of "represent[ing] the class oxen by x and that of horses by y and the conjunction and by the sign + . . . we might represent the aggregate class oxen and horses by x + y".

The logicians' "function" 1850-1950

Eves observes "that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory of sets, or classes, from a foundation in the logic of propositions and propositional functions". But by the late 19th century the logicians' research into the foundations of mathematics was undergoing a major split. The direction of the first group, the Logicists, can probably be summed up best by Bertrand Russell 1903:9 -- "to fulfil two objects, first, to show that all mathematics follows from symbolic logic, and secondly to discover, as far as possible, what are the principles of symbolic logic itself."

The second group of logicians, the set-theorists, emerged with Georg Cantor's "set theory" (18701890) but were driven forward partly as a result of Russell's discovery of a paradox that could be derived from Frege's conception of "function", but also as a reaction against Russell's proposed solution. Zermelo's set-theoretic response was his 1908 Investigations in the foundations of set theory I -- the first axiomatic set theory; here too the notion of "propositional function" plays a role.

George Boole's The Laws of Thought 1854; John Venn's Symbolic Logic 1881

In his An Investigation into the laws of thought Boole now defined a function in terms of a symbol x as follows:

"8. Definition.-- Any algebraic expression involving symbol x is termed a function of x, and may be represented by the abbreviated form f(x)"

Boole then used algebraic expressions to define both algebraic and logical notions, e.g., 1x is logical NOT(x), xy is the logical AND(x,y), x + y is the logical OR(x, y), x(x+y) is xx+xy, and "the special law" xx = x2 = x.

In his 1881 Symbolic Logic Venn was using the words "logical function" and the contemporary symbolism (x = f(y), y = f1(x), cf page xxi) plus the circle-diagrams historically associated with Venn to describe "class relations", the notions "'quantifying' our predicate", "propositions in respect of their extension", "the relation of inclusion and exclusion of two classes to one another", and "propositional function" (all on p. 10), the bar over a variable to indicate not-x (page 43), etc. Indeed he equated unequivocally the notion of "logical function" with "class" [modern "set"]: "... on the view adopted in this book, f(x) never stands for anything but a logical class. It may be a compound class aggregated of many simple classes; it may be a class indicated by certain inverse logical operations, it may be composed of two groups of classes equal to one another, or what is the same thing, their difference declared equal to zero, that is, a logical equation. But however composed or derived, f(x) with us will never be anything else than a general expression for such logical classes of things as may fairly find a place in ordinary Logic".

Frege's Begriffsschrift 1879

Gottlob Frege's Begriffsschrift (1879) preceded Giuseppe Peano (1889), but Peano had no knowledge of Frege 1879 until after he had published his 1889. Both writers strongly influenced Bertrand Russell (1903). Russell in turn influenced much of 20th-century mathematics and logic through his Principia Mathematica (1913) jointly authored with Alfred North Whitehead.

At the outset Frege abandons the traditional "concepts subject and predicate", replacing them with argument and function respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention".

Frege begins his discussion of "function" with an example: Begin with the expression "Hydrogen is lighter than carbon dioxide". Now remove the sign for hydrogen (i.e., the word "hydrogen") and replace it with the sign for oxygen (i.e., the word "oxygen"); this makes a second statement. Do this again (using either statement) and substitute the sign for nitrogen (i.e., the word "nitrogen") and note that "This changes the meaning in such a way that "oxygen" or "nitrogen" enters into the relations in which "hydrogen" stood before". There are three statements:

"Hydrogen is lighter than carbon dioxide." "Oxygen is lighter than carbon dioxide." "Nitrogen is lighter than carbon dioxide."

Now observe in all three a "stable component, representing the totality of [the] relations"; call this the function, i.e.,

"... is lighter than carbon dioxide", is the function.

Frege calls the argument of the function "[t]he sign [e.g., hydrogen, oxygen, or nitrogen], regarded as replaceable by others that denotes the object standing in these relations". He notes that we could have derived the function as "Hydrogen is lighter than . . .." as well, with an argument position on the right; the exact observation is made by Peano. Finally, Frege allows for the case of two (or more arguments). For example, remove "carbon dioxide" to yield the invariant part (the function) as:

"... is lighter than ... "

The one-argument function Frege generalizes into the form (A) where A is the argument and ( ) represents the function, whereas the two-argument function he symbolizes as (A, B) with A and B the arguments and ( , ) the function and cautions that "in general (A, B) differs from (B, A)". Using his unique symbolism he translates for the reader the following symbolism:

"We can read |--- (A) as "A has the property . |--- (A, B) can be translated by "B stands in the relation to A" or "B is a result of an application of the procedure to the object A".

Peano 1889 The Principles of Arithmetic 1889

Peano defined the notion of "function" in a manner somewhat similar to Frege, but without the precision. First Peano defines the sign "K means class, or aggregate of objects", the objects of which satisfy three simple equality-conditions, a = a, (a = b) = (b = a), IF ((a = b) AND (b = c)) THEN (a = c). He then introduces , "a sign or an aggregate of signs such that if x is an object of the class s, the expression x denotes a new object". Peano adds two conditions on these new objects: First, that the three equality-conditions hold for the objects x; secondly, that "if x and y are objects of class s and if x = y, we assume it is possible to deduce x = y". Given all these conditions are met, is a "function presign". Likewise he identifies a "function postsign". For example if is the function presign a+, then x yields a+x, or if is the function postsign +a then x yields x+a.

Bertrand Russell's The Principles of Mathematics 1903

While the influence of Cantor and Peano was paramount, in Appendix A "The Logical and Arithmetical Doctrines of Frege" of The Principles of Mathematics, Russell arrives at a discussion of Frege's notion of function, "...a point in which Frege's work is very important, and requires careful examination". In response to his 1902 exchange of letters with Frege about the contradiction he discovered in Frege's Begriffsschrift Russell tacked this section on at the last moment.

For Russell the bedeviling notion is that of "variable": "6. Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain variables. The notion of the variable is one of the most difficult with which logic has to deal. For the present, I openly wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. . . . We shall find always, in all mathematical propositions, that the words any or some occur; and these words are the marks of a variable and a formal implication".

As expressed by Russell "the process of transforming constants in a proposition into variables leads to what is called generalization, and gives us, as it were, the formal essence of a proposition ... So long as any term in our proposition can be turned into a variable, our proposition can be generalized; and so long as this is possible, it is the business of mathematics to do it"; these generalizations Russell named propositional functions". Indeed he cites and quotes from Frege's Begriffsschrift and presents a vivid example from Frege's 1891 Function und Begriff: That "the essence of the arithmetical function 2*x3+x is what is left when the x is taken away, i.e., in the above instance 2*( )3 + ( ). The argument x does not belong to the function but the two taken together make the whole". Russell agreed with Frege's notion of "function" in one sense: "He regards functions -- and in this I agree with him -- as more fundamental than predicates and

relations" but Russell rejected Frege's "theory of subject and assertion", in particular "he thinks that, if a term a occurs in a proposition, the proposition can always be analysed into a and an assertion about a".

Evolution of Russell's notion of "function" 1908-1913

Russell would carry his ideas forward in his 1908 Mathematical logical as based on the theory of types and into his and Whitehead's 1910-1913 Principia Mathematica. By the time of Principia Mathematica Russell, like Frege, considered the propositional function fundamental: "Propositional functions are the fundamental kind from which the more usual kinds of function, such as sin x or log x or "the father of x" are derived. These derivative functions . . . are called descriptive functions". The functions of propositions . . . are a particular case of propositional functions".

Propositional functions: Because his terminology is different from the contemporary, the reader may be confused by Russell's "propositional function". An example may help. Russell writes a propositional function in its raw form, e.g., as : " is hurt". (Observe the circumflex or "hat" over the variable y). For our example, we will assign just 4 values to the variable : "Bob", "This bird", "Emily the rabbit", and "y". Substitution of one of these values for variable yields a proposition; this proposition is called a "value" of the propositional function. In our example there are four values of the propositional function, e.g., "Bob is hurt", "This bird is hurt", "Emily the rabbit is hurt" and "y is hurt." A proposition, if it is significanti.e., if its truth is determinatehas a truth-value of truth or falsity. If a proposition's truth value is "truth" then the variable's value is said to satisfy the propositional function. Finally, per Russell's definition, "a class [set] is all objects satisfying some propositional function" (p. 23). Note the word "all'" -- this is how the contemporary notions of "For all " and "there exists at least one instance " enter the treatment (p. 15).

To continue the example: Suppose (from outside the mathematics/logic) one determines that the propositions "Bob is hurt" has a truth value of "falsity", "This bird is hurt" has a truth value of "truth", "Emily the rabbit is hurt" has an indeterminate truth value because "Emily the rabbit" doesn't exist, and "y is hurt" is ambiguous as to its truth value because the argument y itself is ambiguous. While the two propositions "Bob is hurt" and "This bird is hurt" are significant (both have truth values), only the value "This bird" of the variable satisfies' the propositional function : " is hurt". When one goes to form the class : : " is hurt", only "This bird" is included, given the four values "Bob", "This bird", "Emily the rabbit" and "y" for variable and their respective truth-values: falsity, truth, indeterminate, ambiguous.

Russell defines functions of propositions with arguments, and truth-functions f(p). For example, suppose one were to form the "function of propositions with arguments" p1: "NOT(p) AND q" and assign its variables the values of p: "Bob is hurt" and q: "This bird is hurt". (We are restricted to the logical linkages NOT, AND, OR and IMPLIES, and we can only assign "significant" propositions to the variables p and q). Then the "function of propositions with arguments" is p1: NOT("Bob is hurt") AND "This bird is hurt"). To

determine the truth value of this "function of propositions with arguments" we submit it to a "truth function", e.g., f(p1): f(NOT("Bob is hurt") AND "This bird is hurt")), which yields a truth value of "truth".

The notion of a "many-one" functional relation": Russell first discusses the notion of "identity", then defines a descriptive function (pages 30ff) as the unique value x that satisfies the (2-variable) propositional function (i.e., "relation") .

N.B. The reader should be warned here that the order of the variables are reversed! y is the independent variable and x is the dependent variable, e.g., x = sin(y).

Russell symbolizes the descriptive function as "the object standing in relation to y": R'y =DEF (x)(x R y). Russell repeats that "R'y is a function of y, but not a propositional function [sic]; we shall call it a descriptive function. All the ordinary functions of mathematics are of this kind. Thus in our notation "sin y" would be written " sin 'y ", and "sin" would stand for the relation sin 'y has to y".

Hardy 1908

Hardy 1908, pp. 2628 defined a function as a relation between two variables x and y such that "to some values of x at any rate correspond values of y." He neither required the function to be defined for all values of x nor to associate each value of x to a single value of y. This broad definition of a function encompasses more relations than are ordinarily considered functions in contemporary mathematics.

The Formalist's "function": David Hilbert's axiomatization of mathematics (1904-1927)

David Hilbert set himself the goal of "formalizing" classical mathematics "as a formal axiomatic theory, and this theory shall be proved to be consistent, i.e., free from contradiction" . In his 1927 The Foundations of Mathematics Hilbert frames the notion of function in terms of the existence of an "object":

13. A(a) --> A((A)) Here (A) stands for an object of which the proposition A(a) certainly holds if it holds of any object at all; let us call the logical -function". [The arrow indicates implies.]

Hilbert then illustrates the three ways how the -function is to be used, firstly as the "for all" and "there exists" notions, secondly to represent the "object of which [a proposition] holds", and lastly how to cast it into the choice function.

Recursion theory and computability: But the unexpected outcome of Hilbert's and his student Bernays's effort was failure. At about the same time, in an effort to solve Hilbert's Entscheidungsproblem, mathematicians set about to define what was meant by an "effectively calculable function" (Alonzo Church 1936), i.e., "effective method" or "algorithm", that is, an explicit, step-by-step procedure that would succeed in computing

a function. Various models for algorithms appeared, in rapid succession, including Church's lambda calculus (1936), Stephen Kleene's -recursive functions(1936) and Allan Turing's (1936-7) notion of replacing human "computers" with utterly-mechanical "computing machines". It was shown that all of these models could compute the same class of computable functions. Church's thesis holds that this class of functions exhausts all the number-theoretic functions that can be calculated by an algorithm. The outcomes of these efforts were vivid demonstrations that, in Turing's words, "there can be no general process for determining whether a given formula U of the functional calculus K [Principia Mathematica] is provable".

Development of the set-theoretic definition of "function"

Set theory began with the work of the logicians with the notion of "class" (modern "set") for example De Morgan (1847), Jevons (1880), Venn 1881, Frege 1879 and Peano (1889). It was given a push by Georg Cantor's attempt to define the infinite in set-theoretic treatment(18701890) and a subsequent discovery of an antinomy (contradiction, paradox) in this treatment (Cantor's paradox), by Russell's discovery (1902) of an antinomy in Frege's 1879 (Russell's paradox), by the discovery of more antinomies in the early 20th century (e.g., the 1897 Burali-Forti paradox and the 1905 Richard paradox), and by resistance to Russell's complex treatment of logic and dislike of his axiom of reducibility (1908, 19101913) that he proposed as a means to evade the antinomies.

Russell's paradox 1902

In 1902 Russell sent a letter to Frege pointing out that Frege's 1879 Begriffsschrift allowed a function to be an argument of itself: "On the other hand, it may also be that the argument is determinate and the function indeterminate . . .." From this unconstrained situation Russell was able to form a paradox:

"You state ... that a function, too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself?"

Frege responded promptly that "Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic".

From this point forward development of the foundations of mathematics became an exercise in how to dodge "Russell's paradox", framed as it was in "the bare [set-theoretic] notions of set and element".

Zermelo's set theory (1908) modified by Skolem (1922)

The notion of "function" appears as Zermelo's axiom IIIthe Axiom of Separation (Axiom der Aussonderung). This axiom constrains us to use a propositional function (x) to "separate" a subset M from a previously formed set M:

"AXIOM III. (Axiom of separation). Whenever the propositional function (x) is definite for all elements of a set M, M possesses a subset M containing as elements precisely those elements x of M for which (x) is true".

As there is no universal setsets originate by way of Axiom II from elements of (non-set) domain B -- "...this disposes of the Russell antinomy so far as we are concerned". But Zermelo's "definite criterion" is imprecise, and is fixed by Weyl, Fraenkel, Skolem, and von Neumann.

In fact Skolem in his 1922 referred to this "definite criterion" or "property" as a "definite proposition":

"... a finite expression constructed from elementary propositions of the form a b or a = b by means of the five operations [logical conjunction, disjunction, negation, universal quantification, and existential quantification].

van Heijenoort summarizes:

"A property is definite in Skolem's sense if it is expressed . . . by a well-formed formula in the simple predicate calculus of first order in which the sole predicate constants are and possibly, =. ... Today an axiomatization of set theory is usually embedded in a logical calculus, and it is Weyl's and Skolem's approach to the formulation of the axiom of separation that is generally adopted.

In this quote the reader may observe a shift in terminology: nowhere is mentioned the notion of "propositional function", but rather one sees the words "formula", "predicate calculus", "predicate", and "logical calculus." This shift in terminology is discussed more in the section that covers "function" in contemporary set theory.

The WienerHausdorffKuratowski "ordered pair" definition 19141921

The history of the notion of "ordered pair" is not clear. As noted above, Frege (1879) proposed an intuitive ordering in his definition of a two-argument function (A, B). Norbert Wiener in his 1914 (see below) observes that his own treatment essentially "revert(s) to Schrder's treatment of a relation as a class of ordered couples". Russell (1903) considered the definition of a relation (such as (A, B)) as a "class of couples" but rejected it:

"There is a temptation to regard a relation as definable in extension as a class of couples. This is the formal advantage that it avoids the necessity for the primitive proposition

asserting that every couple has a relation holding between no other pairs of terms. But it is necessary to give sense to the couple, to distinguish the referent [domain] from the relatum [converse domain]: thus a couple becomes essentially distinct from a class of two terms, and must itself be introduced as a primitive idea. . . . It seems therefore more correct to take an intensional view of relations, and to identify them rather with class-concepts than with classes."

By 1910-1913 and Principia Mathematica Russell had given up on the requirement for an intensional definition of a relation, stating that "mathematics is always concerned with extensions rather than intensions" and "Relations, like classes, are to be taken in extension". To demonstrate the notion of a relation in extension Russell now embraced the notion of ordered couple: "We may regard a relation ... as a class of couples ... the relation determined by (x, y) is the class of couples (x, y) for which (x, y) is true". In a footnote he clarified his notion and arrived at this definition:

"Such a couple has a sense, i.e., the couple (x, y) is different from the couple (y, x) unless x = y. We shall call it a "couple with sense," ... it may also be called an ordered couple.

But he goes on to say that he would not introduce the ordered couples further into his "symbolic treatment"; he proposes his "matrix" and his unpopular axiom of reducibility in their place.

An attempt to solve the problem of the antinomies led Russell to propose his "doctrine of types" in an appendix B of his 1903 The Principles of Mathematics. In a few years he would refine this notion and propose in his 1908 The Theory of Types two axioms of reducibility, the purpose of which were to reduce (single-variable) propositional functions and (dual-variable) relations to a "lower" form (and ultimately into a completely extensional form); he and Alfred North Whitehead would carry this treatment over to Principia Mathematica 1910-1913 with a further refinement called "a matrix". The first axiom is *12.1; the second is *12.11. To quote Wiener the second axiom *12.11 "is involved only in the theory of relations". Both axioms, however, were met with skepticism and resistance. By 1914 Norbert Wiener, using Whitehead and Russell's symbolism, eliminated axiom *12.11 (the "two-variable" (relational) version of the axiom of reducibility) by expressing a relation as an ordered pair "using the null set. At approximately the same time, Hausdorff (1914, p. 32) gave the definition of the ordered pair (a, b) as { {a,1}, {b, 2} }. A few years later Kuratowski (1921) offered a definition that has been widely used ever since, namely { {a, b}, {a} }". As noted by Suppes (1960) "This definition . . . was historically important in reducing the theory of relations to the theory of sets.

Observe that while Wiener "reduced" the relational *12.11 form of the axiom of reducibility he did not reduce nor otherwise change the propositional-function form *12.1; indeed he declared this "essential to the treatment of identity, descriptions, classes and relations".

Schnfinkel's notion of "function" as a many-one "correspondence" 1924

Where exactly the general notion of "function" as a many-one relationship derives from is unclear. Russell in his 1920 Introduction to Mathematical Philosophy states that "It should be observed that all mathematical functions result form one-many [sic -- contemporary usage is many-one] relations . . . Functions in this sense are descriptive functions". A reasonable possibility is the Principia Mathematica notion of "descriptive function" -- R 'y =DEF (x)(x R y): "the singular object that has a relation R to y". Whatever the case, by 1924, Moses Schonfinkel expressed the notion, claiming it to be "well known":

"As is well known, by function we mean in the simplest case a correspondence between the elements of some domain of quantities, the argument domain, and those of a domain of function values ... such that to each argument value there corresponds at most one function value".

According to Willard Quine, Schnfinkel's 1924 "provide[s] for ... the whole sweep of abstract set theory. The crux of the matter is that Schnfinkel lets functions stand as arguments. For Schnfinkel, substantially as for Frege, classes are special sorts of functions. They are propositional functions, functions whose values are truth values. All functions, propositional and otherwise, are for Schnfinkel one-place functions". Remarkably, Schnfinkel reduces all mathematics to an extremely compact functional calculus consisting of only three functions: Constancy, fusion (i.e., composition), and mutual exclusivity. Quine notes that Haskell Curry (1958) carried this work forward "under the head of combinatory logic".

von Neumann's set theory 1925

By 1925 Abraham Fraenkel (1922) and Thoralf Skolem (1922) had amended Zermelo's set theory of 1908. But von Neumann was not convinced that this axiomatization could not lead to the antinomies. So he proposed his own theory, his 1925 An axiomatization of set theory. It explicitly contains a "contemporary", set-theoretic version of the notion of "function":

"[Unlike Zermelo's set theory] [w]e prefer, however, to axiomatize not "set" but "function". The latter notion certainly includes the former. (More precisely, the two notions are completely equivalent, since a function can be regarded as a set of pairs, and a set as a function that can take two values.)".

His axiomatization creates two "domains of objects" called "arguments" (I-objects) and "functions" (II-objects); where they overlap are the "argument functions" (I-II objects). He introduces two "universal two-variable operations" -- (i) the operation [x, y]: ". . . read 'the value of the function x for the argument y) and (ii) the operation (x, y): ". . . (read 'the ordered pair x, y'") whose variables x and y must both be arguments and that itself produces an argument (x,y)". To clarify the function pair he notes that "Instead of f(x) we write [f,x] to indicate that f, just like x, is to be regarded as a variable in this procedure".

And to avoid the "antinomies of naive set theory, in Russell's first of all . . . we must forgo treating certain functions as arguments". He adopts a notion from Zermelo to restrict these "certain functions"

Since 1950

Notion of "function" in contemporary set theory

Both axiomatic and naive forms of Zermelo's set theory as modified by Fraenkel (1922) and Skolem (1922) define "function" as a relation, define a relation as a set of ordered pairs, and define an ordered pair as a set of two "dissymetric" sets.

While the reader of Suppes (1960) Axiomatic Set Theory or Halmos (1970) Naive Set Theory observes the use of function-symbolism in the axiom of separation, e.g., (x) (in Suppes) and S(x) (in Halmos), they will see no mention of "proposition" or even "first order predicate calculus". In their place are "expressions of the object language", "atomic formulae", "primitive formulae", and "atomic sentences".

Kleene 1952 defines the words as follows: "In word languages, a proposition is expressed by a sentence. Then a 'predicate' is expressed by an incomplete sentence or sentence skeleton containing an open place. For example, "___ is a man" expresses a predicate ... The predicate is a propositional function of one variable. Predicates are often called 'properties' ... The predicate calculus will treat of the logic of predicates in this general sense of 'predicate', i.e., as propositional function".

The reason for the disappearance of the words "propositional function" e.g., in Suppes (1960), and Halmos (1970), is explained by Alfred Tarski 1946 together with further explanation of the terminology:

"An expression such as x is an integer, which contains variables and, on replacement of these variables by constants becomes a sentence, is called a SENTENTIAL [i.e., propositional cf his index] FUNCTION. But mathematicians, by the way, are not very fond of this expression, because they use the term "function" with a different meaning. ... sentential functions and sentences composed entirely of mathematical symbols (and not words of everyday languange), such as: x + y = 5 are usually referred to by mathematicians as FORMULAE. In place of "sentential function" we shall sometimes simply say "sentence" --- but only in cases where there is no danger of any misunderstanding".

For his part Tarski calls the relational form of function a "FUNCTIONAL RELATION or simply a FUNCTION" . After a discussion of this "functional relation" he asserts that:

"The concept of a function which we are considering now differs essentially from the concepts of a sentential [propositional] and of a designatory function .... Strictly speaking ... [these] do not belong to the domain of logic or mathematics; they denote certain categories of expressions which serve to compose logical and mathematical statements,

but they do not denote things treated of in those statements... . The term "function" in its new sense, on the other hand, is an expression of a purely logical character; it designates a certain type of things dealt with in logic and mathematics."

Further developments

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. More recently, the concept of functor has been used as an analogue of a function in category theory.

Chapter 2

Inverse Function

A function and its inverse 1. Because maps a to 3, the inverse 1 maps 3 back to a.

In mathematics, if is a function from a set A to a set B, then an inverse function for is a function from B to A, with the property that a round trip (a composition) from A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the function produces an output y, then inputting y into the inverse function produces the output x, and vice versa.

A function that has an inverse is called invertible; the inverse function is then uniquely determined by and is denoted by 1 (read f inverse, not to be confused with exponentiation).

Definitions

If maps X to Y, then 1 maps Y back to X.

Let be a function whose domain is the set X, and whose codomain is the set Y. Then, if it exists, the inverse of is the function 1 with domain Y and codomain X, with the property:

Stated otherwise, a function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inverse function.

Not all functions have an inverse. For this rule to be applicable, each element y Y must correspond to exactly one element x X. This is generally stated as two conditions:

Every corresponds to no more than one ; a function with this property is called one-to-one, or information-preserving, or an injection.

Every corresponds to at least one ; a function with this property is called onto, or a surjection.

A function with both of these properties is called a bijection, so the above is often stated as "a function is bijective if and only if it has an inverse function".

In elementary mathematics, the domain is often assumed to be the real numbers, if not otherwise specified, and the codomain is assumed to be the image. Most functions encountered in elementary calculus do not have an inverse.

Example: squaring and square root functions

The function (x) = x2 may or may not be invertible, depending on the domain and codomain.

If the domain is the real numbers, then each element in Y would correspond to two different elements in X (x), and therefore would not be invertible. More precisely, the square of x is not invertible because it is impossible to deduce from its output the sign of its input. Such a function is called non-injective or information-losing. Notice that neither the square root nor the principal square root function is the inverse of x2 because the first is not single-valued, and the second returns -x when x is negative.

If the domain and codomain are both the non-negative numbers, or if the domain is the negative numbers, then the function is invertible (by the principal square root) and injective.

Inverses in higher mathematics

The definition given above is commonly adopted in calculus. In higher mathematics, the notation

means " is a function mapping elements of a set X to elements of a set Y". The source, X, is called the domain of , and the target, Y, is called the codomain. The codomain contains the range of as a subset, and is considered part of the definition of .

When using codomains, the inverse of a function : X Y is required to have domain Y and codomain X. For the inverse to be defined on all of Y, every element of Y must lie in the range of the function . A function with this property is called onto or a surjection. Thus, a function with a codomain is invertible if and only if it is both one-to-one and onto. Such a function is called a one-to-one correspondence or a bijection, and has the property that every element y Y corresponds to exactly one element x X.

Inverses and composition

If is an invertible function with domain X and range Y, then

This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if Y coincides with the codomain of . Using the composition of functions we can rewrite this statement as follows:

where idX is the identity function on the set X. In category theory, this statement is used as the definition of an inverse morphism.

If we think of composition as a kind of multiplication of functions, this identity says that the inverse of a function is analogous to a multiplicative inverse. This explains the origin of the notation 1.

Note on notation

The superscript notation for inverses can sometimes be confused with other uses of superscripts, especially when dealing with trigonometric and hyperbolic functions. To avoid this confusion, the notations [1] or with the "-1" above the are sometimes used/needed.

It is important to realize that 1(x) is not the same as (x)1. In 1(x), the superscript "1" is not an exponent. A similar notation is used in dynamical systems for iterated functions. For example, 2 denotes two iterations of the function ; if (x) = x + 1, then 2(x) = (x + 1) + 1, or x + 2. In symbols:

In calculus, (n), with parentheses, denotes the nth derivative of a function . For instance:

In trigonometry, for historical reasons, sin2(x) usually does mean the square of sin(x):

However, the expression sin1(x) does not always represent the multiplicative inverse to sin(x). If that is the case then:

Then it denotes the inverse function for sin(x) (actually a partial inverse; see below). To avoid confusion, an inverse trigonometric function is often indicated by the prefix "arc". For instance the inverse sine is typically called the arcsine:

The function (sin x)1 is the multiplicative inverse to the sine, and is called the cosecant. It is usually denoted csc x:

Hyperbolic functions behave similarly, using the prefix "ar", as in arsinh(x), for the inverse function of sinh(x), and csch(x) for the multiplicative inverse of sinh(x).

Properties

Uniqueness

If an inverse function exists for a given function , it is unique: it must be the inverse relation.

Symmetry

There is a symmetry between a function and its inverse. Specifically, if the inverse of is 1, then the inverse of 1 is the original function . In symbols:

This follows because inversion of relations is an involution.

This statement is an obvious consequence of the deduction that for to be invertible it must be injective (first definition of the inverse) or bijective (second definition). The property of symmetry can be concisely expressed by the following formula:

Inverse of a composition

The inverse of g o is 1 o g1.

The inverse of a composition of functions is given by the formula

Notice that the order of and g have been reversed; to undo g followed by , we must first undo and then undo g.

For example, let (x) = x + 5, and let g(x) = 3x. Then the composition o g is the function that first multiplies by three and then adds five:

To reverse this process, we must first subtract five, and then divide by three:

This is the composition (g1 o 1) (y).

Self-inverses

If X is a set, then the identity function on X is its own inverse:

More generally, a function : X X is equal to its own inverse if and only if the composition o is equal to idx. Such a function is called an involution.

Inverses in calculus Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Such functions are often defined through formulas, such as:

A function from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. as long as the graph of the function passes the horizontal line test.

The following table shows several standard functions and their inverses:

Function (x) Inverse 1(y) Notes x + a y a a x a y mx y / m m 0

1 / x 1 / y x, y 0 x2 x, y 0 only x3 no restriction on x and y

xp y1/p (i.e. ) x, y 0 in general, p 0 ex ln y y > 0 ax loga y y > 0 and a > 0

trigonometric functions

inverse trigonometric functions

various restrictions (see table below)

Formula for the inverse

One approach to finding a formula for 1, if it exists, is to solve the equation y = (x) for x. For example, if is the function

then we must solve the equation y = (2x + 8)3 for x:

Thus the inverse function 1 is given by the formula

Sometimes the inverse of a function cannot be expressed by a formula. For example, if is the function

then is one-to-one, and therefore possesses an inverse function 1. There is no simple formula for this inverse, since the equation y = x + sin x cannot be solved algebraically for x.

Graph of the inverse

The graphs of y = (x) and y = 1(x). The dotted line is y = x.

If and 1 are inverses, then the graph of the function

is the same as the graph of the equation

This is identical to the equation y = (x) that defines the graph of , except that the roles of x and y have been reversed. Thus the graph of 1 can be obtained from the graph of by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line y = x.

Inverses and derivatives

A continuous function is one-to-one (and hence invertible) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function

is invertible, since the derivative (x) = 3x2 + 1 is always positive.

If the function is differentiable, then the inverse 1 will be differentiable as long as (x) 0. The derivative of the inverse is given by the inverse function theorem:

If we set x = 1(y), then the formula above can be written

This result follows from the chain rule.

The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiable function : Rn Rn is invertible in a neighborhood of a point p as long as the Jacobian matrix of at p is invertible. In this case, the Jacobian of 1 at (p) is the matrix inverse of the Jacobian of at p.

Real-world examples For example, let be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit:

then its inverse function converts degrees Fahrenheit to degrees Celsius:

Or, suppose assigns each child in a family its birth year. An inverse function would output which child was born in a given year. However, if the family has twins (or triplets) then the output cannot be known when the input is the common birth year. As well, if a year is given in which no child was born then a child cannot be named. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example,

Generalizations

Partial inverses

The square root of x is a partial inverse to (x) = x2.

Even if a function is not one-to-one, it may be possible to define a partial inverse of by restricting the domain. For example, the function

is not one-to-one, since x2 = (x)2. However, the function becomes one-to-one if we restrict to the domain x 0, in which case

(If we instead restrict to the domain x 0, then the inverse is the negative of the square root of x.) Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:

The inverse of this cubic function has three branches.

Sometimes this multivalued inverse is called the full inverse of , and the portions (such as x and x) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch, and its value at y is called the principal value of 1(y).

For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the picture above).

The arcsine is a partial inverse of the sine function.

These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since

for every real x (and more generally sin(x + 2n) = sin(x) for every integer n). However, the sine is one-to-one on the interval [2, 2], and the corresponding partial inverse is called the arcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between 2 and 2. The following table describes the principal branch of each inverse trigonometric function:

function Range of usual principal value sin1 2 sin1(x) 2 cos1 0 cos1(x) tan1 2 < tan1(x) < 2 cot1 0 < cot1(x) < sec1 0 sec1(x) < 2 or 2 < sec1(x) csc1 2 csc1(x) < 0 or 0 < csc1(x) 2

Left and right inverses

If : X Y, a left inverse for (or retraction of ) is a function g: Y X such that

That is, the function g satisfies the rule

Thus, g must equal the inverse of on the range of , but may take any values for elements of Y not in the range. A function has a left inverse if and only if it is injective.

A right inverse for (or section of ) is a function h: Y X such that

That is, the function h satisfies the rule

Thus, h(y) may be any of the elements of x that map to y under . A function has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice).

An inverse which is both a left and right inverse must be unique; otherwise not. Likewise, if g is a left inverse for , then g may or may not be a right inverse for ; and if g is a right inverse for , then g is not necessarily a left inverse for . For example let :R[0,) denote the squaring map, such that (x)=x2 for all x in R, and let g:[0,)R denote the square root map, such that g(x)=x for all x0. Then (g(x))=x for all x in [0,); that is, g is a right inverse to . However, g is not a left inverse to , since, e.g., g((-1))=1-1.

Preimages

If : X Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y Y is the set of all elements of X that map to y:

The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f.

Similarly, if S is any subset of Y, the preimage of S is the set of all elements of X that map to S:

For example, take a function : R R, where : x x2. This function is not invertible for reasons discussed above. Yet preimages may be defined for subsets of the codomain:

The preimage of a single element y Y a singleton set {y} is sometimes called the fiber of y. When Y is the set of real numbers, it is common to refer to 1(y) as a level set.

Chapter 3

Special Functions & Implicit and Explicit Functions

Special functions Special functions are particular mathematical functions which have more or less established names and notations due 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 commonly accepted as special. In particular, elementary functions are also considered as special functions.

Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of 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 efficient algorithms for the evaluation, especially in the complex plane.

Notations used in special functions

In most cases, the standard notation is used for indication of a special function: the name of function, subscripts, if any, open parenthesis, then arguments, separated with comma, and then close parenthesis. Such a notation allows easy translation of the expressions to algorithmic languages avoiding ambiguities. Functions with established international notations are sin, cos, exp, erf, and erfc.

Sometimes, a special function has several names. The natural logarithm can be called as Log, log or ln, depending on the context. For example, the tangent function may be denoted Tan, tan or tg (especially in Russian literature); arctangent may be called atan, arctg, or tan 1. Bessel functions may be written ; usually, ,

, refer to the same function.

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 to confusion.

Superscripts may indicate not only exponentiation, but modification of a function. Examples include:

usually indicates is typically , but never usually means , and not ; this one typically

causes the most confusion, as it is inconsistent with the others.

Evaluation of special functions

Most special functions are considered as a function of a complex variable. They are analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the Taylor or asymptotic series are available. In addition, sometimes there exist relations with other special functions; a complicated special 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 converge slowly if at all. In algorithmic languages, rational approximations are typically used, although they may behave badly in the case of complex argument(s).

History of special functions

Classical theory

While trigonometry can be codified, as was clear already to expert mathematicians of the eighteenth century (if not before), the search for a complete and unified theory of special functions has continued since the nineteenth century. The high point of special function theory in the period 1850-1900 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 the basic 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 trigonometric and exponential functions, was a fundamental tool. The end of the century also saw a very detailed discussion of spherical harmonics.

Changing and fixed motivations

Of course the wish for a broad theory including as many as possible of the known special functions has its intellectual appeal, but it is worth noting other motivations. For a long time, the special functions were in the particular province of applied mathematics; applications to the physical sciences and engineering determined the relative importance of functions. In the days before the electronic computer, the ultimate compliment to a special function was the computation, by hand, of extended tables of its values. This was a capital-intensive process, intended to make the function available by look-up, as for the familiar logarithm tables. The aspects of the theory that then mattered might then 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 other structure behind the faade of endless formulae in rows. There is not a real conflict between these approaches, in fact.

Twentieth century

The twentieth century saw several waves of interest in special function theory. The classic Whittaker and Watson textbook sought to unify the theory by using complex variables; the G. N. Watson tome A Treatise on the Theory of Bessel Functions pushed the techniques as far as possible for one important type that particularly admitted asymptotics to be studied.

The later Bateman manuscript project, under the editorship of Arthur Erdlyi, attempted to be encyclopedic, and came around the time when electronic computation was coming to the fore and tabulation ceased to be the main issue.

Contemporary theories

The modern theory of orthogonal polynomials is of a definite but limited scope. Hypergeometric series became an intricate 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 could be recast in terms of Lie groups. Further, work on algebraic combinatorics also revived interest in older parts of the theory. Conjectures of Ian G. Macdonald helped to open up large and active new fields with the typical special function flavour.

Difference equations have begun to take their place besides differential equations as a source for special functions.

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

Implicit and explicit functions In mathematics, an implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable. To give a function f explicitly is to provide a prescription for determining the output value of the function y in terms of the input value x:

y = f(x).

By contrast, the function is implicit if the value of y is obtained from x by solving an equation of the form:

R(x,y) = 0.

That is, it is defined as the level set of a function in two variables: one variable or the other may determine the other, but one is not given an explicit formula for one in terms of the other.

Implicit functions can often be useful in situations where it is inconvenient to solve explicitly an equation of the form R(x,y) = 0 for y in terms of x. Even if it is possible to rearrange this equation to obtain y as an explicit function f(x), it may not be desirable to do so since the expression of f may be much more complicated than the expression of R. In other situations, the equation R(x,y) = 0 may fail to define a function at all, and rather defines a kind of multiple-valued function. Nevertheless, in many situations, it is still possible to work with implicit functions. Some techniques from calculus, such as differentiation, can be performed with relative ease using implicit differentiation.

The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R(x, y) = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval. Geometrically, the graph defined by R(x,y) = 0 will overlap locally with the graph of a function y = f(x).

Various numerical methods exist for solving the equation R(x,y)=0 to find an approximation to the implicit function y. Many of these methods are iterative in that they produce successively better approximations, so that a prescribed accuracy can be achieved. Many of these iterative methods are based on some form of Newton's method.

Examples

Inverse functions

Implicit functions commonly arise as one way of describing the notion of an inverse function. If f is a function, then the inverse function of f is a solution of the equation

for y in terms of x. Intuitively, an inverse function is obtained from f by interchanging the roles of the dependent and independent variables. Stated another way, the inverse function is the solution y of the equation

R(x,y) = x f(y) = 0.

Examples.

1. The natural logarithm y = ln(x) is the solution of the equation x ey = 0. 2. The product log is an implicit function given by x y ey = 0.

Algebraic functions

An algebraic function is a solution y for an equation R(x,y) = 0 where R is a polynomial of two variables. Algebraic functions play an important role in mathematical analysis and algebraic geometry. A simple example of an algebraic function is given by the unit circle:

x2 + y2 1 = 0.

Solving for y gives

Note that there are two "branches" to the implicit function: one where the sign is positive and the other where it is negative.

Caveats Not every equation R(x, y) = 0 has a graph that is the graph of a function, the circle equation being one prominent example. Another example is an implicit function given by x C(y) = 0 where C is a cubic polynomial having a "hump" in its graph. Thus, for an

implicit function to be a true function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the x-axis and "cutting away" some unwanted function branches. A resulting formula may only then qualify as a legitimate explicit function.

The defining equation R = 0 can also have other pathologies. For example, the implicit equation x = 0 does not define a function at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the domain. The implicit function theorem provides a uniform way of handling these sorts of pathologies.

Implicit differentiation In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.

As explained in the introduction, y can be given as a function of x implicitly rather than explicitly. When we have an equation R(x, y) = 0, we may be able to solve it for y and then differentiate. However, sometimes it is simpler to differentiate R(x, y) with respect to x and then solve for dy/dx.

Examples

1. Consider for example

This function normally can be manipulated by using algebra to change this equation to an explicit function:

Differentiation then gives . Alternatively, one can differentiate the equation:

Solving for :

2. An example of an implicit function, for which implicit differentiation might be easier than attempting to use explicit differentiation, is

In order to differentiate this explicitly with respect to x, one would have to obtain (via algebra)

and then differentiate this function. This creates two derivatives: one for y > 0 and another for y < 0.

One might find it substantially easier to implicitly differentiate the implicit function;

thus,

3. Sometimes standard explicit differentiation cannot be used and, in order to obtain the derivative, another method such as implicit differentiation must be employed. An example of such a case is the implicit function y5 y = x. It is impossible to express y explicitly as a function of x and dy/dx therefore this cannot be found by explicit differentiation. Using the implicit method, dy/dx can be expressed:

facto

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