Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SPECIAL FUNCTIONS

Deduce the value of

J=

∫∞

0

(u+2)^2

(u^2 +4)^5 /^2

du.

18.15 The complex functionz! is defined by

z!=

∫∞

0

uze−udu for Rez>−1.

For Rez≤−1 it is defined by

z!=

(z+n)!

(z+n)(z+n−1)···(z+1)

,

wherenis any (positive) integer>−Rez. Being the ratio of two polynomials,z!

is analytic everywhere in the finite complex plane except at the poles that occur

whenzis a negative integer.

(a) Show that the definition ofz!forRez≤−1 is independent of the value of

nchosen.

(b) Prove that the residue ofz! at the polez=−m,wheremis an integer>0,

is (−1)m−^1 /(m−1)!.

18.16 For− 1 <Rez<1, use the definition and value of the beta function to show
that

z!(−z)! =

∫∞

0

uz

(1 +u)^2

du.

Contour integration gives the value of the integral on the RHS of the above
equation asπzcosecπz. Use this to deduce the value of (−^12 )!.
18.17 The integral

I=

∫∞

−∞

e−k

2

k^2 +a^2

dk, (∗)

in whicha>0, occurs in some statistical mechanics problems. By first considering

the integral

J=

∫∞

0

eiu(k+ia)du,

and a suitable variation of it, show thatI=(π/a)exp(a^2 )erfc(a), where erfc(x)
is the complementary error function.
18.18 Consider two series expansions of the error function as follows.

(a) Obtain a series expansion of the error function erf(x) in ascending powers

ofx. How many terms are needed to give a value correct to four significant

figures for erf(1)?

(b) Obtain an asymptotic expansion thatcan be used to estimate erfc(x)for

largex(>0) in the form of a series

erfc(x)=R(x)=e−x

2 ∑∞

n=0

an

xn

.

Consider what bounds can be put on the estimate and at what point the

infinite series should be terminated in a practical estimate. In particular,

estimate erfc(1) and test the answer for compatibility with that in part (a).

18.19 For the functionsM(a, c;z) that are the solutions of the confluent hypergeometric
equation,