Complex Variablesashwin/Mathematical_Physics... · 2019-12-04 · Complex Variables Problem set: 6...
Transcript of Complex Variablesashwin/Mathematical_Physics... · 2019-12-04 · Complex Variables Problem set: 6...
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.
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