Complex Variables

Problem set: 6

1. Evaluate the following integrals

(a)

∫ ∞0

dx

(x2 + a2)2, a2 > 0

(b)

∫ ∞0

dx

(x2 + a2)(x2 + b2), a2, b2 > 0

2. Evaluate the integral ∫ ∞−∞

x sin x

x2 + a2dx, a2 > 0

3. Consider a rectangular contour CR with corners at (±R, 0) and (±R, a). Show that∮CRe−z

2

dz =

∫ R

−Re−x

2

dx −∫ R

−Re−(x+ia)2dx + JR = 0

where

JR =

∫ a

0

e−(R+iy)2i dy −∫ a

0

e−(−R+iy)2i dy

Show limR→∞ JR = 0, whereupon we have∫∞−∞ e

−(x+ia)2 =∫∞−∞ e

−x2=√π, and consequently,

deduce that∫ +∞−∞ ex

2cos 2ax dx =

√πe−a

2.

4. Evaluate the following definite integral using complex integration

(a)

∫ ∞−∞

eiax

x2 − b2dx, a, b > 0

(b)

∫ ∞0

sin x

x(x2 + 1)dx

Note: The poles may fall on the x axis, therefore choose the contour of integration accordingly.

1

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