Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.13 DEFINITE INTEGRALS USING CONTOUR INTEGRATION

y

−R O R x

Γ

Figure 24.14 A semicircular contour in the upper half-plane.

Evaluate

I=

∫∞

0

dx

(x^2 +a^2 )^4

, whereais real.

The complex function (z^2 +a^2 )−^4 has poles of order 4 atz=±ai,ofwhichonlyz=ai
is in the upper half-plane. Conditions (ii) and (iii) are clearly satisfied. For higher-order
poles, formula (24.56) for evaluating residues can be tiresome to apply. So, instead, we put
z=ai+ξand expand for smallξto obtain§

1

(z^2 +a^2 )^4

=

1

(2aiξ+ξ^2 )^4

=

1

(2aiξ)^4

(

1 −

iξ

2 a

)− 4

.

The coefficient ofξ−^1 is given by

1

(2a)^4

(−4)(−5)(−6)

3!

(

−i

2 a

) 3

=

− 5 i

32 a^7

,

and hence by the residue theorem

∫∞

−∞

dx

(x^2 +a^2 )^4

=

10 π

32 a^7

,

and soI=5π/(32a^7 ).

Condition (i) of the previous method required there to be no poles of the

integrand on the real axis, but in fact simple poles on the real axis can be

accommodated by indenting the contour as shown in figure 24.15. The indentation

at the polez=z 0 is in the form of a semicircleγof radiusρin the upper half-

plane, thus excluding the pole from the interior of the contour.

§This illustrates another useful technique for determining residues.