residues

OPT287, MTH287 Due: Fri. March 7, at 5pm, Wilmot 213 (under the door)

Homework 4 1. Solve the following integrals using residues:

a)

€

dx (x2 + 4)2−∞

∞

∫ ,

b)

€

dx x4 +1−∞

∞

∫ ,

c)

€

dx (x2 + a2)(x2 + b2)−∞

∞

∫ , for a > 0, b > 0,

d)

€

cos(kx)dx (x2 + a2)3−∞

∞

∫ , for k real,

e)

€

eixdx (x2 +1)(x2 + 2x + 2)−∞

∞

∫ ,

f)

€

1 a + sin2θ0

2π

∫ dθ, for a > 1,

g)

€

1 tanθ − 3i0

2π

∫ dθ .

h)

€

sin(kx) x



  

  −∞

∞

∫ 2

dx , for k real.

i)

€

P (x2 + a2)

(x2 + b2)2(x2 −c2)−∞

∞

∫ dx , for a, b, c > 0,

j)

€

P 1

a + cosθ0

2π

∫ dθ , for –1 < a < 1.

In all cases, give predictions to your results before solving the integrals, draw the contours, identify the poles, and simplify your results.