Mathematical Methods for Physicists

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


• The coefficient of–

• The coefficient of

–

0t( ) ( )xHxxH 102 =

( )0≠ntn

( )( )

( ) ( )

( ) ( ) ( )xHxxHxnHn

xHxn

xHxn

xH

nnn

nnn

11

11

22!

2!

2!1

2

+−

+−

=+⇒

=+−


• (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


• 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 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


• 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

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Mathematical Methods for Physicists