Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.13 DEFINITE INTEGRALS USING CONTOUR INTEGRATION

The proof of the lemma is straightforward once it has been observed that, for

0 ≤θ≤π/2,

1 ≥

sinθ

θ

≥

2

π

. (24.71)

Then, since on Γ we have|exp(imz)|=|exp(−mRsinθ)|,

IΓ≤

∫

Γ

|eimzf(z)||dz|≤MR

∫π

0

e−mRsinθdθ=2MR

∫π/ 2

0

e−mRsinθdθ.

Thus, using (24.71),

IΓ≤ 2 MR

∫π/ 2

0

e−mR(2θ/π)dθ=

πM

m

(

1 −e−mR

)

<

πM

m

;

hence, asR→∞,IΓtends to zero sinceMtends to zero.

Find the principal value of

∫∞

−∞

cosmx

x−a

dx, forareal,m> 0.

Consider the function (z−a)−^1 exp(imz); although it has no poles in the upper half-plane
it does have a simple pole atz=a, and further|(z−a)−^1 |→0as|z|→∞. We will use a
contour like that shown in figure 24.15 and apply the residue theorem. Symbolically,
∫a−ρ

−R

+

∫

γ

+

∫R

a+ρ

+

∫

Γ

=0. (24.72)

Now asR→∞andρ→0 we have

∫

Γ→0, by Jordan’s lemma, and from (24.68) and
(24.69) we obtain

P

∫∞

−∞

eimx

x−a

dx−iπa− 1 =0, (24.73)

wherea− 1 is the residue of (z−a)−^1 exp(imz)atz=a, which is exp(ima). Then taking the
real and imaginary parts of (24.73) gives

P

∫∞

−∞

cosmx

x−a

dx=−πsinma, as required,

P

∫∞

−∞

sinmx

x−a

dx=πcosma, as a bonus.

24.13.3 Integrals of multivalued functions

We have discussed briefly some of the properties and difficulties associated with

certain multivalued functions such asz^1 /^2 or Lnz. It was mentioned that one

method of managing such functions is by means of a ‘cut plane’. A similar

technique can be used with advantage to evaluate some kinds of infinite integral

involving real functions for which the corresponding complex functions are multi-

valued. A typical contour employed for functions with a single branch point