Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.13 EXERCISES

(a) use their series representation to prove that

b

d

dz

M(a, c;z)=aM(a+1,c+1;z);

(b) use an integral representation to prove that

M(a, c;z)=ezM(c−a, c;−z).

18.20 The Bessel functionJν(z) can be considered as a special case of the solution
M(a, c;z) of the confluent hypergeometric equation, the connection being

lim

a→∞

M(a, ν+1;−z/a)

Γ(ν+1)

=z−ν/^2 Jν(2

√

z).

Prove this equality by writing each side in terms of an infinite series and showing
that the series are the same.
18.21 Find the differential equation satisfied by the functiony(x) defined by

y(x)=Ax−n

∫x

0

e−ttn−^1 dt≡Ax−nγ(n, x),

and, by comparing it with the confluent hypergeometric function, expressyas
a multiple of the solutionM(a, c;z) of that equation. Determine the value ofA
that makesyequal toM.
18.22 Show, from its definition, that the Bessel function of the second kind, and of
integral orderν, can be written as

Yν(z)=

1

π

[

∂Jμ(z)

∂μ

−(−1)ν

∂J−μ(z)

∂μ

]

μ=ν

.

Using the explicit series expression forJμ(z), show that∂Jμ(z)/∂μcan be written

as

Jν(z)ln

(z

2

)

+g(ν, z),

and deduce thatYν(z) can be expressed as

Yν(z)=

2

π

Jν(z)ln

(z

2

)

+h(ν, z),

whereh(ν, z), likeg(ν, z), is a power series inz.

18.23 Prove two of the properties of the incomplete gamma functionP(a, x^2 ) as follows.

(a) By considering its form for a suitable value ofa, show that the error function

can be expressed as a particular case of the incomplete gamma function.

(b) The Fresnel integrals, of importance in the study of the diffraction of light,

are given by

C(x)=

∫x

0

cos

(π

2

t^2

)

dt, S(x)=

∫x

0

sin

(π

2

t^2

)

dt.

Show that they can be expressed in terms of the error function by

C(x)+iS(x)=Aerf

[√

π

2

(1−i)x

]

,

whereAis a (complex) constant, which you should determine. Hence express

C(x)+iS(x) in terms of the incomplete gamma function.