Mathematical Methods For Physicists Arfken Pdf 5th Edition

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Mathematical Methods for Physicists

by 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

() ()

∑

∞

+−

txt

xHetxg

)

()

12016032

124816

128

+−=

−=

()

12072048064

246

−+−=

xxxxH

xxxH

xxH

txty 2

+−=

∑

∞

xxxxH

Recurrence Relations (1/4)

•(2)

()

)

xnHxxHxH

nnn 11

−+

)

()

() () ()

()

∑∑∑

∑∑

∑

∞

−

∞

−

∞

−+−

∞

+−

−

=+−⇒

−

=+−⇒

−

=+−⇒

⎟

⎠

⎞

⎜

⎝

⎛

!1!

txt

xtx

extx

Recurrence Relations (2/4)

• The coefficient of

–

• The coefficient of

–

()

)

xHxxH

)

0 ≠ nt

()

)

() () ()

xHxxHxnH

nnn

+−

=+⇒

−

Recurrence Relations (3/4)

•(3)

– Differentiate the generating function with respect

()

)

xnHxH

−

)

()

() ()

∑∑

∑

∞∞

∞

+−

∞

+−

′

=⇒

′

=⇒

′

=⇒

⎟

⎠

⎞

⎜

⎝

⎛

txt

Recurrence Relations (4/4)

• The coefficient of

–

)

()

′

0 = n

0 > n

)

()

)

() ()

xnHxH

!!1

−

′

⇒

−

Value at

)

()

() ( )

()

00:12

10:2

!2!1

242

=+=

−==

=−⇒

−=−+−=+

−

+=⇒

∞

−

+−

∑∑

∑

txt

Hkn

ttttt

etxg

Parity Relation

•

– Expand the generating function

– We have

)()

)

xHxH

−−= 1

)

() () ()

() () () ()

()() ()() ()

() () () () ()

() () () ()() ()

() () () () () ()

xHxxxxH

xHxxxH

xHxxH

6246

535

424

112072048064

112016032

1124816

1128

124

−=−−+−−−=

−=−+−−−=

−=+−−−=

−=−−−=

−=−−=−

−=−=−

)

Rodrigues Representation of

z Differentiation of the generating function times

with respect to (note that )

z Set

()

∑

∞

−−+−+−+−

====

22222

xtxxxtxttxt

xHeeeetxg

() ( )

)

exH

−

−=⇒

0 = t

() ()

xtxt

−−−−

−=

Calculus of Residues

• Multiply the generating function by

• Integrate around the origin

• We have

1 −− m

()

∫

+−−−

= dtet

txtm

Series Form

•

() ( )

()

()()

[]

()

()()

()

[]

∑

−

−−

−

−=

−⋅⋅⋅⋅

⎟

⎠

⎞

⎜

⎝

⎛

−=

+⋅⋅

−

⋅

−

−=

!!2

12531

312

!4!4

!2!2

sns

nnn

ssn

xxH

Orthogonality

• By recurrence relations,

• Let

– We have (4)

– which is self-adjoint and orthogonal in

()

)

022

'''

=+− xnHxxHxH

nnn

()

)

xHex

−

)

012

2''

=−++ xxnx

)

, x

Normalization

• Multiply (1) by itself and by

• Integrate from to , and consider the

orthogonal property

• Equating coefficients of like powers of to obtain

() ()

∑

∞

−+−+−−

2222

xtxtsxsx

xHxHeeee

−

()

[]

()

∑

∫∫

∑

∫

∞

∞−

−−−

∞

∞−

+−+−−

∞

∞−

−

222

2222

sttsxtxtsxsx

dxeedxedxxHe

ππ

()

[]

ndxxHe

∫

∞

−

∞−

Simple Harmonic Oscillator

•(5)

• Reduce to the form of

• Which is (4) with

•Hence

() ()

zEKzx

ψψ

=+∇−

)

()

=−+ xx

ψλ

()

)

xHenx

412

−

−−

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()("

+ xnyyxxxy

/(1 )

()

2(1)

xz z

xdz

izz

−−

−

/(1 )

( )

2(1)

xz z

yx dz

izz

−−

′

=−

−

/(1 )

( )

2(1)

xz z

yx dz

izz

−−

−

′′

=−

−

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's

equation

() ()

()

xz z

xxn

edz

izz

zz zz

−−

−

⎡⎤

−

−+

⎢⎥

−

−−

⎢⎥

⎣⎦

∫ v

/(1 )

2(1)

xz z

idz zz

−−

⎡⎤

⎢⎥

−

⎣⎦

∫ v

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 the

term in the series remains.

• Identify as the generating function for the Lagurre

polynomials.

)

/(1 )

()

2(1)

xz z

Lx dz

izz

−−

−

∫

∑

∞

−−

−

)1(

)(

),(

zxz

zxL

zxg

1 < z

1 −− n

1 −

()

zxg ,

Rodrigues' Formula

• With the transformation

, ,

• Which is the new contour enclosing in the

-plane

• By Chauchy's integral theorem (for derivatives)

(integral ) (9)

−=

)(

xLn

−

()

∫

−

= ds

snx

Series Form

• From these representations of , we find the series

form for integral

(10)

• We have

)

() ()

()

() 1 !

!1! 2!

(1) (1)

( )!( )! ! ( )!( )! !

nn n

mns

Lx x x x n

nx nx

nmmm nsnss

−−

−

⎤

−−

=−+ −+−

⎥

⎦

=− =−

−−−

∑∑

)

()

43 2

2! 4 2

3! 9 18 6

4! 16 72 96 24

Lx x

Lx x x

Lx x x x

Lx 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

)1/()]()()1[()()(2

)()()12()()1(

+−+−=

−−+=+

−−

−+

nxLxLxxLxL

xnLxLxnxLn

nnn

)()()(

xnLxnLxxL

nnn −

−=

)

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 is

which can be verified by using the generating function

• The orthonormal function satisfies the differential

equation

– which has the Sturm-Liouville form (self-adjoint).

)()(

xLex

−

∞≤

x 0

nmnm

dxxLxLe

)()(

∫

∞

−

)

0)()4/2/1()()(

−

′′

xxnxxx

Associated Laguerre Polynomials

• Associated generating functions - Laguerre

polynomial

• 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 obtain

and

)]([)1()( xL

n +

−=

)

() 1

Lx =

1)(

++−= kxxL

(2)(1)

() ( 2)

xkk

Lx k x

=−+ + "

()!

() (1) ( 1)

()!()!!

Lx k

nmkmm

−>−

−+

∑

)!(

)0(

zxz

zxL

)(

)1(

)1/(

∑

∞

−−

−

Recurrence Relations

• Recurrence relations can easily be derived from the

generating function or by differentiating the Laguerre

polynomial recurrence relations

–

)()()()12()()1(

xLknxLxknxLn

n −+

+−−++=+

)()()()(

xLknxnLxxL

n −

−−=

Associated Laguerre Equation

• Differentiating the Laguerre's differential equation

times, we have the associated Laguerre equation

–(11)

0)()()1()(

=+−+= xnLxLxkxxL

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 .

)(

knx

nkx

+−

−

()

)

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

−

∫

∞

−

)!(

)()(

dxxLxLxe

)

xLxex

kxk

22 −

)

()

)

0 42124

2'''

=−+++−++ xxkknxxxx

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

kxk

+−

∫

∞

+−

)12(

)!(

)()( kn

dxxLxLxe

)

1 −

)

xkn 212

()

)

0 42124

2''

=−+++−+ xxkknxx

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

=−∇−

)

ϕθ

)

ERR

+−

⎟

⎠

⎞

⎜

⎝

⎛

−

)1(

Hydrogen Atom (2/4)

• By use of abbreviations , ,and

• (13) becomes

(14)

•where

()

)

0)(

)1(

1)(1

⎥

⎦

⎤

⎢

⎣

⎡

−−+

⎥

⎦

⎤

⎢

⎣

⎡

ρχ

ρρ

ρχ

ρρ

)0(

<−= E

mZe

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

()

)()(

12/

ρρρρχ

−−

+−

12 + L

1, 2, 3, n

2 hn

meZ

−=

Hydrogen Atom (4/4)

• By the result of , we have ,

• With the Bohr radius

• We have the normalized hydrogen wave function

a =

31/2

/2 2 1

2( 1)!

(, , ) ( ) ( ) (, )

2( )!

rLL M

nLM nL L

ZnL

rerLrY

na n n L

θϕ α α θϕ

−+

−−

⎡⎛ ⎞ ⎤

−−

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦

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

21/2

() , 1

(1 2 ) ( 1/ 2)!

Txt t

xt t

∞

−+ −

∑

0 =

21 ±=

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 −=

)

12 !

−

∑

∞

−−

+−

−

12/1

)(

221

txnT

txt

t 2

∑

∞

−

+−

−

2/1

)(2

txnT

txt

Chebyshev Polynomials of Type I (2/2)

• For , define

• Then (16)

• For , define to preserve the

recurrence relation

0 > n

)(

2/1

xnTxT

−

∑

∞

+−

−

)(21

txT

txt

0 = n

1)(

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 +=

∑

∞

+−

2/12/1

2/1

)(

)21(

txT

txt

)()(

2/1

xUxT

∑

∞

+−

)(

txU

txt

)

−

+− 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)(

−+

xTxxTxT

nnn

0)()(2)(

−+

xUxxUxU

nnn

Special Values

0)0(

)1()0(

)1()1(

1)1(

−=

−=−

+ n

0)0(

)1()0(

)1()1(

1)1(

−=

−=−

+ n

Parity and Rodrigue's Representations

• Parity relation

–and

• Rodrigue's representations

–

)()1()( xTxT

−−=

)()1()( xUxU

−−=

])1[(

)!2/1(2

)1()1(

)(

2/12

2/122/1

−

−−

])1[(

)1()!2/1(2

)1()1(

)(

2/12

2/121

2/12/1

−

−+

+−

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(

xnTxnxTxTx

nnn −

+−=−

)()1()()()1(

xUnxnxUxUx

nnn −

++−=−

0)()()()1(

2'''2

=+−− xTnxxTxTx

0)()2()(3)()1(

'''2

=++−− xUnnxxUxUx

0)12(')1(2)1(

''2

=++++−− ynnxyyx

ββ

Power Series Representation

•Define

• From the generating function, or the differential

equations

–

•

• Finally, we obtain

)(1)(

xUxxV

−=

mnm

mnn

]2/[

)2(

)!2(!

)!1(

)1(

)(

−

∑

−

−−

−=

mnm

]2/[

)2(

)!2(!

)!(

)1()(

−

∑

−

−=

21 32 522

13 5

() 1 (1 ) (1 )

nn nn nn

Vx x x x x x x

−− −

⎤

=− − − + − −

⎦

,])1([)()(

2/12 n

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

()

)

−

−= xxw

()

)

1 wx x =−

21/2

() ()(1 ) 2, 0

, 0

TxTx x dx mn

−

≠

⎧

⎪

==≠

⎨

⎪

⎩

∫

21/2

() ()(1 ) 2, 0

0, 0

VxVx x dx mn

−

≠

⎧

⎪

==≠

⎨

⎪

⎩

∫

21/2

() ()(1 )

mn mn

UxUx x dx

−

−=

∫

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 1

Tx x ≤−≤≤

]

1,1

)

xnxT

coscos

−

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

(1)(1)

() (,,;) 1

1! ( 1) !

abx aa bb x

yx Fabcx

cccn

⋅++

==++

0, 1, x =∞

1 < x

bac

1 −

> 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()2)(1()(

−

=−+++=

naaaaa

1)(

()()

(,,;)

() !

Fabcx

∞

∑

Representation of Elementary Functions

• Many elementary functions can be represented by the

hypergeometric equations

• For example

–

– complete elliptic integrals

);2,1,1()1ln(

xFxx

);1,2/1,2/1(

)sin1(

2/122

kFdkK −=−=

∫

θθ

);1,2/1,2/1(

)sin1(

2/122

kFdkE −=−=

∫

θθ

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

( ) ( 1 , 1 ,2 ; ), 2,3,4,

yx x Fa cb c cx c

−

=+−+−−≠ "

Alternate Forms

• Alternate forms of hypergeometric equation include

)]21()1[(

)1(

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

−

−++−++−

⎟

⎠

⎞

⎜

⎝

⎛

−

aby

cbazba

0)(4)(

)122()()1(

222

=−

⎥

⎦

⎤

⎢

⎣

⎡

−

+++−− zabyzy

zbazy

Contiguous Function Relations

• We expect recurrence relations involving unit

changes in the parameters , , and .

• Usual nomenclature for the hypergeometric

functions in which one parameter changes by or

is "contiguous function"

• For example,

22 2

21 21

(){( 1)1 [()1](1))}(,,;)

()( 1)(1,1,;) )( 1)(1,1,;)

abcab a b ab x Fabcx

caab bFa b cx caab aFa b cx

−+−+−−+−−−

= − −+ − + +− −+ + −

Hypergeometric Representations (1/2)

• Gegenbauer function,

–

• Legendre and associate Legendre functions

–

– Alternate forms are

)

;1,12,(

!!2

)!2(

)(

nnF

+++−

ββ

)

;1,1,()(

nnFxP

+−=

)

;1,1,(

)1(

)!(

)(

mnmnmF

−

+++−

−

221

(2 )!

( ) ( 1) ( , 1/2,1/2; )

2!!

(2 1)!

( 1) ( , 1/2,1/2;

(2 )!!

Px Fnn x

nn x

=− − +

−

=− − +

21 21

(2 1)!

( ) ( 1) ( , 3/2,3/2; )

2!!

(2 1)!

( 1) ( , 3/2,3/2; )

(2 )!!

xxFnnx

Fnn x

=− − +

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,,()(

nnFxT

−=

)

;2/3,2,()1()(

nnFnxU

−

+−+=

)

;2/3,1,1(1)(

nnFnxxV

++−−=

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

= x

Solutions

• One solution of the confluent hypergeometric

equation 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

–

(1)

() (,;) (,;) 1

1! ( 1) !

ax aa x

yx Facx Macx

cccn

==++ +

!)(

)(

);,(

xcaM

∑

∞

" ,4,3,2),;1,()(

≠−+=

−

cxaaMxxy

(,;) ( 1 ,2 ;)

(,;)

Macx x Ma c cx

Uacx

−

sin ( )!( 1)! ( 1)!(1 )! cacc a c

⎤

+− −

=−

⎥

−− −−

⎦

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(

)(

2/1

xxMdtex

ππ

∫

−

() ( )

xaaMxadttexa

−+==

−−−

∫

;1,,

}

0 > a

Integral Representation (1/2)

• Confluent hypergeometric functions in integral forms

–,

Re Re

– , Re Re

∫

−−−

−

−ΓΓ

)1(

)()(

)(

);,( dttte

caa

xcaM

acaxt

∫

∞

−−−−

−

)1(

)(

);,( dttte

xcaU

acaxt

}

> c

}

0 > a

}

> x

}

0 > a

Integral Representation (2/2)

• Three important techniques for deriving or verfying

integral 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

++−=

+−

µµ

0)(

4/1

)(

⎟

⎠

⎞

⎜

⎝

⎛

−

++−+ xM

µµ

);12,2/1()(

2/12/

xkUxexW

++−=

+−

µµ

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

−−=

() ( 1/2,2 1;2 )

Jx Mv v ix

−

⎛⎞

=++

⎜⎟

⎝⎠

() ( 1/2,2 1;2)

Ix Mv v x

−

⎛⎞

=++

⎜⎟

⎝⎠

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

221

(2 )! 2(2 1)!

() (1) ( ,1/2; ), () (1) ( ,3/2; )

H x xM n x H x xM n x

=− − =− −

);1,()( xnMxL

)

);1,(

)!(

)()1()( xmnM

+−

=−=

Use of Hypergeometric Funtions

• Expressing special functions in terms of

hypergeometric and confluent hypergeometric

functions 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

... The underlying reason for this importance is that many of the special functions of mathematical physics can be expressed in terms of confluent hypergeometric functions and many of the differential equations of physics, chemistry, and engineering can be reduced to the confluent hypergeometric equation and thus solved in terms of confluent hypergeometric functions. This is particularly true for quantum mechanics [1,2,3,4,5,6,7,8,9,10,11,12], where, for example, the bound state problems for the simple harmonic oscillator in one, two and three dimensions, the Coulomb problem in two and three dimensions, and the Cartesian onedimensional Morse potential can all be solved in terms of confluent hypergeometric functions. In addition, continuum problems, such as the free particle in one, two, and three dimensions, the linear potential, the continuum of the Coulomb problem in two and three dimensions, and the continuum of the Cartesian one-dimensional Morse potential, can also be solved using confluent hypergeometric functions. ...

... Moreover, we note that there is a very nice discussion of Landau levels that also employs confluent hypergeometric functions [13]. The confluent hypergeometric equation also arises in optics [10,14,15,16,17], classical electrodynamics [1,14,18], classical waves [2,19,20], diffusion [21], fluid flow [22], heat transfer [23], general relativity [24,25,26,27], semiclassical quantum mechanics [28], quantum chemistry [29,30], graphic design [31], and many other areas. The solutions of the confluent hypergeometric equation depend in an essential way on whether or not a, b, and a − b are integers and the standard references (see below) do not present these solutions, with appropriate qualifications, in a user-friendly way. ...

... Our principal references on the confluent hypergeometric functions are the NIST Digital Library of Mathematical Functions (DLMF) [33], the precursor volume, The Handbook of Mathematical Functions, by Milton Abramowitz and Irene A. Stegun (AS) [34], Confluent Hypergeometric Functions, by L. J. Slater [35], and Higher Transcendental Functions, edited by Arthur Erdélyi [36]. Some other useful sources of information about confluent hypergeometric functions are Mathematical Methods for Physicists, by George B. Arfken, Hans J. Weber, and Frank E. Harris [10], Methods of Theoretical Physics, by Philip M. Morse and Herman Feshbach [2], A Course of Modern Analysis, by E. T. Whittaker and G. N. Watson [37], Special Functions in Physics with MATLAB, by Wolfgang Schweizer [38],the Wolfram MathWorld website [39,40,41,42,43], and a beautiful dynamic calculator of the Kummer Function, M(a, b, z), and the Tricomi function, U(a, b, z), on the Wolfram website [44]. There is also a Wikipedia entry titled "Confluent hypergeometric functions" [45]. ...

Jr. Wesley N. Mathews
Mark A. Esrick
ZuYao Teoh
James K. Freericks

The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and z^{1-b}M(1+a-b,2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. All three of these functions must be considered in a search for two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve physics problems. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. Our results are summarized in Table I in Section 3. As an example, we use these solutions to study the bound states of the hydrogenic atom, going beyond the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physics instruction that involves the confluent hypergeometric differential equation.

... where R ¼ ÀY À1 W, G ¼ Y À1 Z: Now, assuming that F(t) is the fundamental matrix (see Appendix III) corresponding to the homogeneous part _ C ¼ RC of Eq. (11) and using the initial condition given by Eq. (2), the solution (see (Das and Verma 2014;Raisinghania 2014;Arfken et al., 2012;Ross 2014;Hartman 2010)) of Eq. (11) is given by (12) Eq. (12) gives the interface-concentrations C i 's, i ¼ 1, 2, :::, s: Finally the values of C i 's are used in Eq. (6) to obtain the drug-concentration profiles C ðiÞ ðr, tÞ in the corresponding layers and the release medium. ...

... If f 1 , f 2 , … , f n are the eigenvalues (Das and Verma 2014;Raisinghania 2014;Arfken et al., 2012;Ross 2014 ;Hartman 2010) of the matrix W and v 1 , v 2 , … , v n are the corresponding eigenvectors respectively, then the fundamental matrix, F(t) (see (Das and Verma 2014;Raisinghania 2014;Arfken et al., 2012;Ross 2014 ;Hartman 2010)) corresponding to the homogeneous system _ CðtÞ ¼ WCðtÞ, is given as: FðtÞ ¼ v 1 e f 1 t v 2 e f 2 t : : : v n e f n t Â Ã ...

Saqib Mubarak
M. A. Khanday

In this paper, two mathematical models have been formulated by extending the basic reaction–diffusion model, along with suitable initial and boundary conditions to study the drug delivery and its diffusion in biological tissues from multi-layered capsules/tablets and other drug delivery devices (DDDs), respectively. These devices are either taken orally or through other drug-administration routes. The formulated models are solved using the variational finite element method followed by the fundamental matrix method, to study the drug delivery and its diffusion more efficiently. The main aim of this work is to provide an effective model, using optimal mathematical techniques to help researchers and biologists in medicine in decreasing the endeavours and expenses in designing DDDs. The outcomes obtained are compared with the experimental data to demonstrate the validity and the feasibility of the proposed work.

... A function,f , on S 2 can also be approximated using the complex version of the orthonormal spherical harmonic basis functions, Y m n (θ, φ), [7] as followŝ ...

... 2. Addition theorem. The addition theorem for spherical harmonics [25,7] is given by ...

Ali Gurbuz

This dissertation is on the development of a computational tool to simulate the red blood cell (RBC) flow in unbounded and vessel domains using boundary integral methods. The motivation behind this study is to investigate the role of the mechanosensitive ion channels (MSC) found in the RBC membrane by simulating the motion and deformation of the RBC under various flow conditions. The MSCs have an open-closed state driven by the mechanical stresses in the cell membrane to regulate the ion flow into and out of cells. Recent studies have shown that mutations of the PIEZO1, a type of MSC found in the RBC membrane, are linked to hereditary anemia. Although distinct roles have been identified for the PIEZO1 in the RBC membrane, the coherent understanding of its function is currently unknown. A computational tool that can be used to study the mechanics of RBCs in unbounded and vessel domains might lead to an important insight into the PIEZO1 channel activity and its role in the function of the RBCs. This work focuses on the computational aspects of the RBC flow simulations. The hydrodynamics of the cytoplasm and extracellular plasma fluid is described as an incompressible and viscous flow. In developing a mathematical model, the hydrodynamics of the viscous fluid is presented by weakly singular boundary integral equations. The thin shell mechanics formulations are developed for the RBC membrane mechanics. In these formulations, the in-plane membrane viscoelasticity is modeled using the standard linear solid model. The cell membrane is discretized globally using the spherical harmonic basis functions. This discretization amounts to the spectral Galerkin boundary integral method for the solution of the BIEs describing the motion and deformation of the RBCs. The high-order weakly singular boundary element method is used for the solution procedure for the velocity and traction fields on the vessel surface. Special care is taken on the computation of the weakly and nearly singular integrals on the cell surface, and vessel surface since the accuracy of the flow simulations depends on the accurate computation of these singular integrals. The numerical examples consist of a single RBC's and multiple RBCs' flows in an unbounded domain under the influence of shear flow, and a pressure-driven RBC flow in straight and constricted vessels are presented to demonstrate the capacity of the computational tool created in this work. The effects of the viscoelasticity on the motion and deformation of the RBCs are investigated in these examples.

... with Y m l being the spherical harmonics [32], and R 1 (r ) and R 2 (r ) must satisfy, ⎛ ...

... is associated Laguerre polynomial [32]. Using (9) we obtain, ...

M. D. de Oliveira
Alexandre Grezzi de Miranda Schmidt

We study the modified Dirac oscillator with spin and pseudo-spin symmetries in deformed nuclei. For that, we consider the Dirac equation in curved space-time which line element is \(ds^2 = (1+\alpha ^2 U(r))^2(dt^2- dr^2) - r^2d\theta ^2-r^2\sin ^2\theta d\phi ^2\) and electromagnetic field \(A_{\mu } = (V(r), cA_r(r),0,0)\). From this coupling of \( A_{\mu } \) with the curved space-time emerged two symmetries: Spin symmetry with \( V (r) = U (r) \) and pseudo-spin symmetry with \(V(r)= -U(r)\). In both cases, we solve the Dirac equation exactly and get the energy spectrum. We also analyze the probability density and some energy spectra for both cases.

... .8) being the Gell-Mann matrices. In this case, the generators of the SU(3) group can be written as [45] ...

Guang-Ping Chen
Pu Tu
Chang-Bing Qiao
Xiao-Fei Zhang

We consider a harmonically trapped rotating spin-1 Bose–Einstein condensate with SU(3) spin–orbit coupling subject to a gradient magnetic field. The effects of SU(3) spin–orbit coupling, rotation, and gradient magnetic field on the ground-state structure of the system are investigated in detail. Our results show that the interplay among SU(3) spin–orbit coupling, rotation, and gradient magnetic field can result in a variety of ground states, such as a vortex ring and clover-type structure. The numerical results agree well with our variational analysis results.

... Our preferred form of the Fourier transforms as in [Foll,7.2,7.5], [ArfWeb,20.2]: ...

Alexander Figotin

Coupled-cavity traveling wave tube (CCTWT) is a high power microwave (HPM) vacuum electronic device used to amplify radio-frequency (RF) signals. CCTWTS have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in CCTWTs take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Operational features of a CCTWT particularly the amplification mechanism are similar to those of a multicavity klystron (MCK). We advance here a Lagrangian field theory of CCTWTs with the space being represented by one-dimensional continuum. The theory integrates into it the space-charge effects including the so-called debunching (electron-to-electron repulsion). The corresponding Euler-Lagrange equations are ODEs with coefficients varying periodically in the space. Utilizing the system periodicity we develop the instrumental features of the Floquet theory including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include in particular the dispersion relations and the frequency dependent gain foundational to the RF signal amplification. Serpentine (folded, corrugated) traveling wave tubes are very similar to CCTWTs and our theory applies to them also.

... The methods in this work utilize standard QMT [23,24] and standard mathematical methods [25] showing that there is no sign change of the standard commutator relations when transforming from a laboratory-fixed to a molecule-attached coordinate system. Consistent application of standard AM algebra in the establishment of computed spectra yield nice agreement with laboratory experimental results [5] and agreement in analysis of astrophysical C 2 Swan data from the white dwarf Procyon B [5], including agreement in comparisons with computed spectra that are obtained with other molecular fitting programs such as PGOPHER [26]. ...

Christian Parigger

The interpretation of optical spectra requires thorough comprehension of quantum mechanics, especially understanding the concept of angular momentum operators. Suppose now that a transformation from laboratory-fixed to molecule-attached coordinates, by invoking the correspondence principle, induces reversed angular momentum operator identities. However, the foundations of quantum mechanics and the mathematical implementation of specific symmetries assert that reversal of motion or time reversal includes complex conjugation as part of anti-unitary operation. Quantum theory contraindicates sign changes of the fundamental angular momentum algebra. Reversed angular momentum sign changes are of heuristic nature and are actually not needed in analysis of diatomic spectra. This work addresses sustenance of usual angular momentum theory, including presentation of straightforward proofs leading to falsification of the occurrence of reversed angular momentum identities. This review also summarises aspects of a consistent implementation of quantum mechanics for spectroscopy with selected diatomic molecules of interest in astrophysics and in engineering applications.

... Table 1에 제시하였다. 2차원 무한 매질에서 음향파 파동 방정식의 해석해는 다 음과 같다 (Arfken et al., 2012). (Alford et al., 1974;Cohen, 2002;Zauderer, 2006;Min et al., 2016;Ikelle and Amundsen, 2018) 안정성 결과만 제시하면 Table 4와 같다 (Lines et al., 1999). ...

Insightful knowledge on quantum nanostructured materials is paramount to engineer and exploit their vast gamut of applications. Here, a formalism based on the single-band effective mass equation was developed to determine the light absorption of colloidal quantum dots (CQDs) embedded in a wider bandgap semiconductor host, employing only three parameters (dots/host potential barrier, effective mass, and QD size). It was ascertained how to tune such parameters to design the energy level structure and consequent optical response. Our findings show that the CQD size has the biggest effect on the number and energy of the confined levels, while the potential barrier causes a linear shift of their values. While smaller QDs allow wider energetic separation between levels (as desired for most quantum-based technologies), the larger dots with higher number of levels are those that exhibit the strongest absorption. Nevertheless, it was unprecedently shown that such quantum-enabled absorption coefficients can reach the levels (10 4-10 5 cm −1) of bulk semiconductors.

Zamir Ben-Hur
David Lou Alon
Or Berebi
Boaz Rafaely

Binaural reproduction of high-quality spatial sound has gained considerable interest with the recent technology developments in virtual and augmented reality. The reproduction of binaural signals in the Spherical-Harmonics (SH) domain using Ambisonics is now a well-established methodology, with flexible binaural processing realized using SH representations of the sound-field and the Head-Related Transfer Function (HRTF). However, in most practical cases, the binaural reproduction is order-limited, which introduces truncation errors that have a detrimental effect on the perception of the reproduced signals, mainly due to the truncation of the HRTF. Recently, it has been shown that manipulating the HRTF phase component, by ear-alignment, significantly reduces its effective SH order while preserving its phase information, which may be beneficial for alleviating the above detrimental effect. Incorporating the ear-aligned HRTF into the binaural reproduction process has been suggested by using Bilateral Ambisonics, which is an Ambisonics representation of the sound-field formulated at the two ears. While this method imposes challenges on acquiring the sound-field, and specifically, on applying head-rotations, it leads to a significant reduction in errors caused by the limited-order reproduction, which yields a substantial improvement in the perceived binaural reproduction quality even with first order SH.

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