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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