Quiz 2
EP 2110 - Introduction to Mathematical Physics
Course Instructor: Ashwin Joy
Time duration: 50 minsTotal Marks: 20Electronic devices such as smart-phones, calculators are not allowed.
Important Theorems
Cauchy-Riemann conditions :∂u
∂x=∂v
∂y;∂v
∂x= −∂u
∂y,
provided f(z) = u(x, y) + i v(x, y) is analytic at some z = x+ i y
Cauchy’s Theorem:∮C
f(z) dz = 0, provided f(z) analytic everywhere in D bounded by C
Cauchy’s Integral Formula:
f(z) =1
2πi
∮C
f(ζ)
ζ − zdζ, provided f(z) analytic everywhere in D bounded by C
Cauchy’s Residue Theorem:1
2πi
∮C
f(z) dz =n∑
j=1
Residue[zj],
provided f(z) analytic everywhere in D except at the n singular points zj enclosed C
Laurent series :
f(z) =∞∑
n=−∞
Cn(z − z0)n for some R1 < |z − z0| < R2 and Cn =1
2πi
∮C
f(ζ)
(ζ − z0)n+1dζ
Do any four of the following questions. Each question carries 5 marks.
1. If the real part of an analytic function f(z) is given as xy, write f(z) in terms of z.
2. Evaluate1
2πi
∮C
cosh z
(2 ln 2− z)5dz where C is the circle |z| = 2.
3. Find the Laurent series for the functionez
z2 − 1about z = 1 and give the residue at this point.
4. Evaluate1
2πi
∮C
z + 1
2z3 − 3z2 − 2zdz where C is the circle |z| = 1.
5. Evaluate
∫ ∞0
cos x
1 + x2dx by any method of your choice.
Hint: If you use complex contour integration, this becomes a walk in the park!
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