Mathematical Methods for Physicistsby G. Arfken
Chapter 13: Special Functions
Reporters:黃才哲、許育豪
Hermite Functions
• Generating functions - Hermite polynomial• Recurrence relation• Special values of Hermite polynomial• Alternate representations• Orthogonality• Normalization • Application
Generating Functions
• Define (1)
• Take– expand
– We have
( ) ( )∑∞
=
+− ==0
2
! ,
2
n
n
ntxt
ntxHetxg
( )( )( )( )( )( ) 12016032
124816
128
24
21
35
244
23
22
1
0
+−=
+−=
−=
−=
==
( ) 12072048064 2466
5
−+−=
xxxxH
xxxH
xxxH
xxH
xxHxH
txty 22 +−=
∑∞
=
=0 !n
ny
nye
xxxxH
Recurrence Relations (1/4)
• (2) ( ) ( ) ( )xnHxxHxH nnn 11 22 −+ +=( )
( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )( )∑∑∑
∑∑
∑
∑
∞
=
−∞
=
∞
=
+
∞
=
−∞
=
∞
=
−+−
∞
=
+−
−=+−⇒
−=+−⇒
−=+−⇒
⎟⎠
⎞⎜⎝
⎛=
1
1
00
1
0
1
0
0
12
0
2
!1!2
!2
!1!22
!122
!2
2
n
nn
n
nn
n
nn
n
nn
n
nn
n
nntxt
n
nntxt
tn
xHtn
xHxtn
xH
tn
xHtn
xHxtx
tn
xHextx
tn
xHedtd
Recurrence Relations (2/4)
• The coefficient of–
• The coefficient of
–
0t( ) ( )xHxxH 102 =
( )0≠ntn
( )( )
( ) ( )
( ) ( ) ( )xHxxHxnHn
xHxn
xHxn
xH
nnn
nnn
11
11
22!
2!
2!1
2
+−
+−
=+⇒
=+−
Recurrence Relations (3/4)
• (3)– Differentiate the generating function with respect
to
( ) ( )xnHxH nn 12 −=′
( )
( )
( ) ( )
( ) ( )∑∑
∑∑
∑
∑
∞∞+
∞
=
∞
=
∞
=
+−
∞
=
+−
′=⇒
′=⇒
′=⇒
⎟⎠
⎞⎜⎝
⎛=
1
00
0
2
0
2
2
!!2
!2
!2
2
nnnn
n
nn
n
nn
n
nntxt
n
nntxt
txHtxH
tn
xHtn
xHt
tn
xHte
tn
xHedxdx
== 00 !! nn nn
Recurrence Relations (4/4)
• The coefficient of
–
–
( ) ( ) 0!0 0
0 =′=′
xHxH
nt
0=n
0>n ( )( )
( )
( ) ( )xnHxHn
xHn
xH
nn
nn
1
1
2!!1
2
−
−
=′⇒
′=
−
Value at 0=x
( )( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) 00:12!
!210:2
!!1
!1
!2!11
!2!11
,
12
2
00
20
242222
2
2
2
=+=
−==
=−⇒
−=−+−=+−
+−
+=⇒
=
+
∞
=
∞
=
∞
=
−
+−
∑∑
∑
k
kk
n
n
nk
kk
k
kkt
txt
HknkkHkn
ntxH
kt
kttttte
etxg
Parity Relation
•– Expand the generating function– We have
( ) ( ) ( )xHxH nk
n −−= 1
( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )xHxxxxH
xHxxxxH
xHxxxH
xHxxxH
xHxxH
xHxxH
xH
66246
6
5535
5
4424
4
332
3
222
2
11
1
0
112072048064
112016032
1124816
1128
124
12
1
−=−−+−−−=
−=−+−−−=
−=+−−−=
−=−−−=
−=−−=−
−=−=−
=−
( )xHnRodrigues Representation of
Differentiation of the generating function times with respect to (note that )
Set
( ) ( ) ( )∑∞
=
−−+−+−+− ====0
22
! ,
222222
n
n
nxtxxxtxttxt
ntxHeeeetxg
( ) ( ) ( )22
1 xn
nxn
n edxdexH −−=⇒
0=t
( ) ( )22 xtxt edxde
dtd −−−− −=
nt
Calculus of Residues
• Multiply the generating function by• Integrate around the origin• We have
1−−mt
( ) ∫ +−−−= dteti
mxH txtmm
21 2
2!π
Series Form
• ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )[ ]
( )
( ) ( ) ( )[ ]
∑
∑
=
−
=
−
−−
−−=
−⋅⋅⋅⋅⎟⎟⎠
⎞⎜⎜⎝
⎛−=
+⋅⋅−⋅
+−⋅
−=
2
0
2
2
0
2
22
!!2!22
125312
22
312!4!4
!42!2!2
!22
n
s
sns
n
s
sns
nnnn
ssnnx
ss
nx
xn
nxn
nxxH
Orthogonality
• By recurrence relations,
• Let – We have (4)– which is self-adjoint and orthogonal in
( ) ( ) ( ) 022 ''' =+− xnHxxHxH nnn
( ) ( )xHex nx
n22−=ψ
( ) ( ) ( ) 012 2'' =−++ xxnx nn ψψ( )∞∞−∈ ,x
Normalization
• Multiply (1) by itself and by
• Integrate from to , and consider the orthogonal property
• Equating coefficients of like powers of to obtain
x( ) ( )∑
∞
=
−+−+−− =0,
22
!!2222
nm
nm
nmxtxtsxsx
nmtsxHxHeeee
2xe−
∞− ∞
( ) ( )[ ] ( )
( )∑
∫∫∑ ∫∞
=
∞
∞−
−−−∞
∞−
+−+−−∞
=
∞
∞−
−
==
==
0
21
221
222
0
2
!2
!!22222
n
nnst
sttsxtxtsxsx
nn
xn
nste
dxeedxedxxHenn
st
ππ
st( )[ ] !2 2
122
ndxxHe nn
x π=∫∞
−
∞−
Simple Harmonic Oscillator
• (5)• Reduce to the form of• Which is (4) with • Hence
( ) ( )zEKzxm
ψψ =+∇− 22
21
2 ( ) ( ) ( ) 022
2
=−+ xxdx
xd ψλψ
12 += nλ
( ) ( ) ( )xHenx nxn
n221412 2
!2 −−−−= πψ
Laguerre Functions
• Laguerre polynomial • Associated Laguerre polynomials • Application
Laguerre Polynomial
• Laguerre’s differential equation• Generating functions - Laguerre polynomial• Alternate representations - Rodrigues’ formula• Recurrence relation• Orthogonality
Laguerre’s Differential Equation (1/2)
• (6)• Denote solution as , since will depend on .• By contour integral,
(7)
• The contour encloses the origin but does not enclose the point , since
,
0)(')1()(" =+−+ xnyyxxxyyny n
/(1 )
1
1( ) 2 (1 )
xz z
n ny x dzi z z
eπ
− −
+=−
1=z/(1 )
2
1( ) 2 (1 )
xz z
n ny x dzi z z
eπ
− −
′ = −−∫
/(1 )
3 1
1( ) 2 (1 )
xz z
n ny x dzi z z
eπ
− −
−′′ = −
−∫
∫
Laguerre’s Differential Equation (2/2)
• Substituting into the left-hand side of (6), we obtain
– Which is equal to
• If integrate around a contour so that the final point equals to the initial point, the integral will vanish, thus verifying that (7) is a solution to the Laguerre’sequation
( ) ( ) ( )( )1
3 2 11
1 1 2 11 1
xz znn n
x x n e dzi z zz z z zπ
− −+−
⎡ ⎤−− +⎢ ⎥
−− −⎢ ⎥⎣ ⎦∫
/(1 )1
2 (1 )
xz z
n
d dzi dz z z
eπ
− −⎡ ⎤⎢ ⎥−⎣ ⎦
∫
Generating Functions
• Define the Lagurre polynomial , by
– This is exactly what we would obtain from the series
, (8)
• If multiply by and integrate around the origin, only theterm in the series remains.
• Identify as the generating function for the Lagurrepolynomials.
( )xLn/(1 )
1
1( )2 (1 )
xz z
n nL x dzi z z
eπ
− −
+=−∫
∑∞
=
−−
=−
=0
)1(
)(1
),(n
nn
zxz
zxLz
ezxg 1<z
1−−nz1−z
( )zxg ,
Rodrigues’ Formula
• With the transformation
, ,
• Which is the new contour enclosing in the -plane
• By Chauchy’s integral theorem (for derivatives)
(integral ) (9)
xsz
xz−=
−1 sxsz −
=
xs =
n)(!
)( xnn
nx
exdxdexLn
−=η
s
( )( )∫ +
−
−= ds
xses
iexL n
snx
n 12π
Series Form
• From these representations of , we find the series form for integral
(10)
• We have
( )xLn
n( ) ( ) ( )
2221 2
0 0
1 1( ) 1 !
! 1! 2!
! !( 1) ( 1)( )!( )! ! ( )!( )! !
n
nnn n n
m n sn nm n s
m s
n nnL x x x x nn
n x n xn m m m n s n s s
− −
−−
= =
⎡ ⎤− −= − + − + −⎢ ⎥
⎢ ⎥⎣ ⎦
= − = −− − −∑ ∑( )( )( )( )( )
0
1
22
3 23
4 3 24
1
1
2! 4 2
3! 9 18 6
4! 16 72 96 24
L x
L x x
L x x x
L x x x x
L x x x x x
=
= − +
= − +
= − + − +
= − + − +
Recurrence Relations
• Differentiate the generating function, with respect to and ,
• For reasons of numerical stability, these are used for machine computation of numerical values of .
• The computing machine starts with known numerical values of and .
xz
)1/()]()()1[()()(2)()()12()()1(
11
11
+−+−=−−+=+
−−
−+
nxLxLxxLxLxnLxLxnxLn
nn nn
nnn
)()()( 1' xnLxnLxxL nnn −−=
( )xLn
( )xL0( )xL1
-
-
Orthogonality
• The Laguerre differential equation is not self-adjoint and the Laguerre polynomials do not by themselves form an orthogonal set
• The related set of function is orthonormal for interval ,that iswhich can be verified by using the generating function
• The orthonormal function satisfies the differential equation
– which has the Sturm-Liouville form (self-adjoint).
)()( 2/ xLex nx
n−=ϕ
∞≤≤ x0nmnm
x dxxLxLe ,0)()( δ=∫
∞ −
( )xnϕ
0)()4/2/1()()( =−++′+′′ xxnxxx ϕϕϕ
Associated Laguerre Polynomials
• Associated generating functions - Laguerrepolynomial
• Associated recurrence relation• Associated Laguerre’s differential equation • Alternate representations - associated Rodrigues’
formula• Associated orthogonality
Associated Laguerre Polynomials
• Associated Laguerre polynomials– From the series form of– , ,
– In general, ,• A generating function may be developed by
differentiating the Laguerre generating function times
• Adjusting the index to , we obtainand
)]([)1()( xLdxdxL knk
kkk
n +−=( )xLn
0 ( ) 1kL x = 1)(1 ++−= kxxLk2
2( 2)( 1)( ) ( 2)
2 2k x k kL x k x + +
= − + +
0
( )!( ) ( 1) ( 1)( )!( )! !
nk mn
m
n kL x kn m k m m=
+= − > −
− +∑
k
knL +
!!)!()0(
knknLk
n+
=
n
n
knk
zxz
zxLz
e )()1( 0
1
)1/(
∑∞
=+
−−
=−
Recurrence Relations
• Recurrence relations can easily be derived from the generating function or by differentiating the Laguerrepolynomial recurrence relations.
––
)()()()12()()1( 11 xLknxLxknxLn kn
kn
kn −+ +−−++=+
)()()()( 1'' xLknxnLxxL k
nkn
kn −−−=
Associated Laguerre Equation
• Differentiating the Laguerre’s differential equation times, we have the associated Laguerre equation– (11)0)()()1()( '' =+−+= xnLxLxkxxL k
nkn
kn
k
Associated Rodrigues Representation
• A Rodrigues representation of the associated Laguerre polynomial is–
• Note that all of these formula reduce to the corresponding expressions for when .
)(!
)( knxn
nkxkn xe
dxd
nxexL +−−
=
( )xLkn
( )xLn 0=k
Self-Adjoint (1/2)
• The associated Laguerre equation is not self-adjoint, but it can be put in self-adjoint form by multiplying
– We obtain
– Let , satisfies the self-adjoint equation
kxxe−
∫∞ − +
=0 ,!
)!()()( nmkm
kn
kx
nkndxxLxLxe δ
( ) ( )xLxex kn
kxkn
22−=ψ ( )xknψ
( ) ( ) ( )[ ] ( ) 0 42124 2''' =−+++−++ xxkknxxxx kn
kn
kn ψψψ
Self-Adjoint (2/2)
• Define• Substitution into the associated Laguerre equation
yields (12)– The corresponding normalization integral is
– It shows that do not form an orthogonal set (except with as the weighting function) because of the term
)()( 2/)1(2/ xLxex kn
kxkn
+−=φ
∫∞ +− ++
+=
0
1 )12(!
)!()()( knn
kndxxLxLxe kn
kn
kx
( ){ }xknφ1−x
( ) xkn 212 ++
( ) ( )[ ] ( ) 0 42124 2'' =−+++−+ xxkknxx kn
kn φφ
Hydrogen Atom (1/4)• The solution of the Schödinger wave equation
–
• The angular dependence of is • The radial part , satisfies the equation
– (13)
ψψψ Er
Zem
h=−∇−
22
2
2 ψ ( )ϕθ ,MLY
( )rRERR
rLL
mhR
rZe
drdRr
drd
rmh
=+
+−⎟⎠⎞
⎜⎝⎛− 2
222
2
2 )1(2
12
Hydrogen Atom (2/4)
• By use of abbreviations , ,and
• (13) becomes
(14) • where ( ) ( )αρρχ R=
0)()1(41)(1
22
2 =⎥⎦
⎤⎢⎣
⎡ +−−+⎥
⎦
⎤⎢⎣
⎡ρχ
ρρλ
ρρχρ
ρρLL
dd
dd
rαρ = )0(82
2 <−= EhmEα
2
22h
mZeα
λ =
Hydrogen Atom (3/4)
• A comparison with (12) for shows that (14) is satisfied by– In which is replaced by and
• Since the Laguerre function of nonintegral would diverge as , must be an integer
• The restriction on has the effect on quantizing the energy
( )xknφ
)()( 121
12/ ρρρρχ λρ +
−−+−= L
LL Le
k 12 +L 1−− Lλn
ρρ en λ 1,2,3,n=λ
22
42
2 hnmeZEn −=
Hydrogen Atom (4/4)
• By the result of , we have ,• With the Bohr radius• We have the normalized hydrogen wave function
nE rna
Z
0
2=ρ
02
2 22na
ZnZ
hme
==α
2
2
0 meha =
3 1/2/2 2 1
10
2 ( 1)!( , , ) ( ) ( ) ( , )2 ( )!
r L L MnLM n L L
Z n Lr e r L r Yna n n L
αψ θ ϕ α α θ ϕ− +− −
⎡ ⎛ ⎞ ⎤− −= ⎜ ⎟⎢ ⎥+⎣ ⎝ ⎠ ⎦
Chebyshev Polynomials
• Chebyshev polynomials• Generating function• Recurrence relations• Special values• Parity relation• Rodrigue’s representations• Recurrence relations – derivatives• Power series representation• Orthogonality• Numerical applications
Generating Function
• The generating function for the ultraspherical or Gegenbauer polynomials
(15)
– gives rise to the Legendre polynomials– , generate two sets of polynomials known
as the Chebyshev polynomials
1/ 2
2 1/ 20
2 ( ) , 1(1 2 ) ( 1/ 2)!
nn
n
T x t txt t
ββ
β
πβ
∞
+=
= <− + − ∑
0=β
21±=β
Chebyshev Polynomials of Type I (1/2)
• With ,the and dependence on the left of (15) disappears and the blows up
• To avoid the problem,– differentiate (15) with respect to and let
to yield
• Then multiply and add one to obtain
21−=β t x( )1 2 !β −
t
∑∞
=
−−=+−
−
0
12/12 )(
221 n
nn txnT
txttx π
t2
∑∞
=
−+=+−
−
0
2/12
2
)(22
1211
n
nn txnT
txtt π
21−=β
Chebyshev Polynomials of Type I (2/2)
• For , define
• Then (16)
• For , define to preserve the recurrence relation
0>n )(2
)( 2/1 xnTxT nn−=
π
∑∞
=
+=+−
−
02
2
)(21211
n
nn txT
txtt
0=n 1)(0 =xT
Chebyshev Polynomials of Type II
• With , (15) becomes– Define – This gives us
(17)– The functions generated by
are called the Chebyshev polynomials of type II
21+=β ∑∞
=
=+− 0
2/12/12
2/1
)()21(
2n
nn txT
txtπ
)()(2
2/1 xUxT nn =π
∑∞
=
=+− 0
2 )(21
1n
nn txU
txt( )xUn ( ) 1221 −
+− ttx
Recurrence Relations
• From generating functions (16) and (17), we obtain (18)(19)
• Then, use the generating functions for the first few values of and these recurrence relations to obtain the high-order polynomials
0)()(2)( 11 =+− −+ xTxxTxT nnn
0)()(2)( 11 =+− −+ xUxxUxU nnn
n
Special Values
0)0()1()0(
)1()1(
1)1(
12
2
=−=
−=−
=
+n
nn
nn
n
TT
T
T
0)0()1()0(
)1()1(
1)1(
12
2
=−=
−=−
=
+n
nn
nn
n
UU
U
U
Parity and Rodrigue’s Representations
• Parity relation– and
• Rodrigue’s representations
–
–
)()1()( xTxT nn
n −−= )()1()( xUxU nn
n −−=
])1[()!2/1(2
)1()1()( 2/122/122/1
−−−
−−= n
n
n
n
n
n xdxd
nxxT π
])1[()1()!2/1(2
)1()1()( 2/122/121
2/12/1+
+ −−++−
= nn
n
n
n
n xdxd
xnnxU π
Recurrence Relations – Derivatives
• From the generating functions, obtain a variety of recurrence relations involving derivatives––
• From (18) and (19)Type I satisfies (20)Type II satisfies (21)
• The Gegenbauer’s equation–– which is a generalization of these equations
)()()()1( 1'2 xnTxnxTxTx nnn −+−=−
)()1()()()1( 1'2 xUnxnxUxUx nnn −++−=−
0)()()()1( 2'''2 =+−− xTnxxTxTx nn n
0)()2()(3)()1( '''2 =++−− xUnnxxUxUx nn n
0)12(')1(2)1( ''2 =++++−− ynnxyyx ββ
Power Series Representation
• Define • From the generating function, or the differential
equations–
–
•• Finally, we obtain
)(1)( 21 xUxxV nn −=+
mnn
m
mn x
mnmmnnxT 2
]2/[
)2()!2(!)!1()1(
2)( −∑ −
−−−=
mnn
m
mn x
mnmmnxU 2
]2/[
)2()!2(!
)!()1()( −∑ −−
−=
( ) ( ) ( )2 1 3 2 5 2 21 3 5( ) 1 (1 ) (1 )n n n n n n
nV x x x x x x x− − −⎡ ⎤= − − − + − −⎣ ⎦
,])1([)()( 2/12 nnn xixxiVxT −+=+ 1≤x
Orthogonality
• If (20) and (21) are put into self-adjoint form, we obtain and as their weighting factors
• The resulting orthogonality integrals are
( ) ( ) 2121 −−= xxw ( ) ( )1 221w x x= −
1 2 1/ 2
1
0, ( ) ( )(1 ) 2, 0
, 0m n
m nT x T x x dx m n
m nππ
−
−
≠⎧⎪− = = ≠⎨⎪ = =⎩
∫
1 2 1/ 2
1
0, ( ) ( )(1 ) 2, 0
0, 0m n
m nV x V x x dx m n
m nπ−
−
≠⎧⎪− = = ≠⎨⎪ = =⎩
∫
1 2 1 / 2,1
( ) ( ) (1 )2m n m nU x U x x d x π δ
−− =∫
Numerical Applications
• The Chebyshev polynomials are useful in numerical work over an interval because–– The maxima and minima are of comparable
magnitude– The maxima and minima are spread reasonably
uniformly over the range – These properties follow from
[ ]1,1−
( ) 1, 1 1nT x x≤ − ≤ ≤
[ ]1,1−( ) ( )xnxTn
1coscos −=
Hypergeometric Functions
• Hypergeometric equations• Contiguous function relations• Hypergeometric representations
Hypergeometric Equations
• Hypergeometric equations–– A canonical form of a linear second-order differential
equation with regular singularities at .
• One solution is
– Which is known as the hypergeometric equation or hypergeometric series
– The range of convergence: for , and , for
0)()(])1([)()1( =−′++−+′′− xabyxyxbacxyxx
2
2 1( 1) ( 1)( ) ( , , ; ) 1
1! ( 1) !a b x a a b b xy x F a b c xc c c n⋅ + +
= = + ++
0, 1, x = ∞
1<x1=x
bac +>1−=x1−+> bac
Pochhammer Symbol
• In terms of the Pochhammer symbol,– and– The hypergeometric equations becomes
– The leading subscripts 2 indicates that two Pochhammer symbols appear in the numerator and the final subscript 1 indicates one Pochhammer symbol in the denominator
)!1()!1()1()2)(1()(
−−+
=−+++=a
nanaaaaa n 1)( 0 =a
2 10
( ) ( )( , , ; )( ) !
nn n
n n
a b xF a b c xc n
∞
=
= ∑
Representation of Elementary Functions
• Many elementary functions can be represented by the hypergeometric equations
• For example–– complete elliptic integrals
);2,1,1()1ln( 12 xFxx −=+
);1,2/1,2/1(2
)sin1( 212
2/
0
2/122 kFdkK −=−= ∫πθθ
π
);1,2/1,2/1(2
)sin1( 212
2/
0
2/122 kFdkE −=−= ∫πθθ
π
Hypergeometric Equations
• Another solution–– It shows that if is an integer, either the two
solutions coincide or one of the solutions will blow up.
– In such case the second solution is expected to include a logarithmic term
12 1( ) ( 1 , 1 ,2 ; ), 2,3, 4,cy x x F a c b c c x c−= + − + − − ≠
c
Alternate Forms
• Alternate forms of hypergeometric equation include
02
12
1)]21()1[(2
1)1( 2
22 =⎟
⎠⎞
⎜⎝⎛ −
−⎟⎠⎞
⎜⎝⎛ −
−++−++−⎟⎠⎞
⎜⎝⎛ −
−zabyzy
dzdcbazbazy
dzdz
0)(4)(21)122()()1( 2222
22 =−⎥⎦
⎤⎢⎣⎡ −
+++−− zabyzydzd
zczbazy
dzdz
Contiguous Function Relations
• We expect recurrence relations involving unit changes in the parameters , , and .
• Usual nomenclature for the hypergeometricfunctions in which one parameter changes by or is “contiguous function”
• For example,
a b c
1+ 1−
2 2 22 1
2 1 2 1
( ){ ( 1) 1 [( ) 1](1 ))} ( , , ; )( )( 1) ( 1, 1, ; ) )( 1) ( 1, 1, ; )a b c a b a b a b x F a b c xc a a b b F a b c x c a a b a F a b c x− + − + − − + − − −
= − − + − + + − − + + −
Hypergeometric Representations (1/2)
• Gegenbauer function, –
• Legendre and associate Legendre functions–
–– Alternate forms are
)2
1;1,12,(!!2)!2()( 12
xnnFn
nxTn−
+++−+
= ββββ
ββ
)2
1;1,1,()( 12xnnFxPn
−+−=
)2
1;1,1,(!2
)1()!()!()( 12
22 xmnmnmFm
xmnmnxP m
mm
n−
+++−−
−+
=−
22 2 12
22 1
(2 )!( ) ( 1) ( , 1/2,1/2; )2 ! !(2 1)! ( 1) ( , 1/2,1/2; )(2 )!!
nn n
n
nP x F n n xn nn F n n xn
= − − +
−= − − +
22 1 2 12
22 1
(2 1)!( ) ( 1) ( , 3/2,3/2; )2 ! !(2 1)! ( 1) ( , 3/2,3/2; )(2 )!!
nn n
n
nP x x F n n xn n
n x F n n xn
+
+= − − +
+= − − +
Hypergeometric Representations (2/2)
• In terms of hypergeometric functions, the Chebyshev functions become–––
• The leading factors are determined by direct comparison of complete power series, comparison of coefficients of particular powers of the variable, or evaluation at or , etc
)2
1;2/1,,()( 12xnnFxTn
−−=
)2
1;2/3,2,()1()( 12xnnFnxUn
−+−+=
)2
1;2/3,1,1(1)( 122 xnnFnxxVn
−++−−=
0=x 1=x
Confluent Hypergeometric Functions
• Confluent hypergeometric equation• Confluent hypergeometric representations• Integral representation• Bessel and modified Bessel functions• Hermite functions• Miscellaneous cases
Confluent Hypergeometric Equation
•
• May be obtained from the hypergeometric equation by merging two of its singularities
• The resulting equation has a regular singularity at and an irregular one at .
0)()()()( =−′−+′′ xayxyxcxyx
0=x ∞=x
Solutions
• One solution of the confluent hypergeometricequation is – Which is convergent for all finite – In terms of the Pochhammer symbols, we have
which becomes a polynomial if the parameter is or a negative integer
• A second solution is• The standard form of the Confluent hypergeometric
equation is a linear combination of both solutions–
2
1 1( 1)( ) ( , ; ) ( , ; ) 1
1! ( 1) !a x a a xy x F a c x M a c xc c c n
+= = = + + +
+x
!)()();,(
0 nx
caxcaM
n
n n
n∑∞
=
=
a 1,4,3,2),;1,()( 1 ≠−+= − cxaaMxxy c
1( , ; ) ( 1 , 2 ; )( , ; )cM a c x x M a c c xU a c x π −
sin ( )!( 1)! ( 1)!(1 )!c a c c a cπ⎡ ⎤+ − −
= −⎢ ⎥− − − −⎣ ⎦
Representations
• Numerous elementary functions may be represented by the confluent hypergeometric function
• For example– Error function erf– Incomplete gamma function
, Re
);2/3,2/1(22)( 22/102/1
2
xxMdtexx t
ππ ∫ == −
( ) ( )xaaMxadttexa ax at −+== −−−∫ ;1,, 1
0
1γ { } 0>a
Integral Representation (1/2)
• Confluent hypergeometric functions in integral forms – ,
Re Re
– , Re Re
∫ −−− −−ΓΓ
Γ=
1
0
11 )1()()(
)();,( dtttecaa
cxcaM acaxt
∫∞ −−−− −
Γ=
0
11 )1()(
1);,( dtttea
xcaU acaxt
{ }>c { } 0>a
{ }>x { } 0>a
Integral Representation (2/2)
• Three important techniques for deriving or verfyingintegral representations:– Transformation of generating functions and
Rodrigues representations– Direct integration to yield a series– Verification that the integral representation
satisfies the differential equation, exclusion of other solution, verification of normalization
Self-Adjoint
• The confluent hypergeometric equation is not self-adjoint.
• Define – This new function is a Whittaker function which
satisfies the self-adjoint equation
– The corresponding second solution is
);12,2/1()( 2/12/ xkMxexM xk ++−= +− µµµµ
0)(4/141)( 2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛ −++−+ xM
xxkxM k
nk µµ
µ
);12,2/1()( 2/12/ xkUxexW xk ++−= +− µµµµ
Bessel and Modified Bessel Functions
• Kummer’s first formula is useful in representing Bessel and modified Bessel functions
• Representation in the form of the confluent hypergeometric equation– Bessel function– The modified Bessel functions of the first kind
);,();,( xcacMexcaM x −−=
( ) ( 1/ 2,2 1;2 )! 2
vix
ve xJ x M v v ixv
− ⎛ ⎞= + +⎜ ⎟
⎝ ⎠
( ) ( 1/ 2, 2 1;2 )! 2
vx
ve xI x M v v xv
− ⎛ ⎞= + +⎜ ⎟
⎝ ⎠
Hermite Functions
• The Hermite functions are given by
• Comparing the Laguerre differential equation with the confluent hypergeometric equation, we have
– The constant is fixed as unity, since• The associated Laguerre functions
– Alternate verification is obtain by comparing with the power series solution
2 22 2 1
(2 )! 2(2 1)!( ) ( 1) ( ,1/ 2; ), ( ) ( 1) ( ,3 / 2; )! !
n nn n
n nH x xM n x H x xM n xn n+
+= − − = − −
);1,()( xnMxLn −=
( ) 10 =nL
);1,(!!)!()()1()( xmnM
mnmnxL
dxdxL mnm
mmm
n +−+
=−= +
Use of Hypergeometric Funtions
• Expressing special functions in terms of hypergeometric and confluent hypergeometricfunctions let the behavior of the special functions follows as a series of special cases
• This may be useful in determining asymptotic behavior or evaluating normalization integrals
• The relations between the special functions are clarified