Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX VARIABLES

y

−R O R x

Γ

γ

Figure 24.15 An indented contour used when the integrand has a simple

pole on the real axis.

What is then obtained from a contour integration, apart from the contributions

for Γ andγ, is called theprincipal value of the integral,defined asρ→0by

P

∫R

−R

f(x)dx≡

∫z 0 −ρ

−R

f(x)dx+

∫R

z 0 +ρ

f(x)dx.

The remainder of the calculation goes through as before, but the contribution

from the semicircle,γ, must be included. Result (24.63) of section 24.12 shows

that since only a simple pole is involved its contribution is

−ia− 1 π, (24.69)

wherea− 1 is the residue at the pole and the minus sign arises becauseγis

traversed in the clockwise (negative) sense.

We defer giving an example of an indented contour until we have established

Jordan’s lemma; we will then work through an example illustrating both. Jordan’s

lemma enables infinite integrals involving sinusoidal functions to be evaluated.

For a functionf(z)of a complex variablez,if

(i)f(z)is analytic in the upper half-plane except for a finite number of poles inImz> 0 ,

(ii)the maximum of|f(z)|→ 0 as|z|→∞in the upper half-plane,

(iii)m> 0 ,

then

IΓ=

∫

Γ

eimzf(z)dz→ 0 asR→∞, (24.70)

whereΓis the same semicircular contour as in figure 24.14.

Note that this condition (ii) is less stringent than the earlier condition (ii) (see the

start of this section), since we now only requireM(R)→0 and notRM(R)→0,

whereMis the maximum§of|f(z)|on|z|=R.

§More strictly, the least upper bound.