Question

Evaluate the following integral considering closed curves are positively oriented. The problem may involve Cauchy’s theorem, the integral formulas, and/or the deformation theorem. ∫ γ f(z)dz Given: f(z)= z^4 / z−2i ; γ is any closed path enclosing 2i.

Answer 1

1. Use Cauchy Integral Formula to evaluate the following integrals (C stands for a simple closed curve counterclockwise oriented): e sin

(a) fidz, where C : [2]= 6. (Answer: 2 je sin 5)

(b) Scd, where C: []=2 and n=0,1,2,3,... T

(Answer: 0 if n = 0, 2πj if n = 1, 4πj if n = 2,2πj if n = 3,0 if n ≥ 4). (c) Jedz, where is the rectangle with vertices -2,2.-2-3j. 2-3j.

(Answer: ).

2:41 (d) Sc dz, where C: |=|=2.

(Answer: 0).

2. Use Cauchy Integral Formula for derivatives to evaluate J where = 1 is oriented counterclockwise, a € R and n is a positive integer. 2 Use your result to evaluate the integral east cos(asint-nt)dt. (Answer: The first integral is 2" and the second is 2").

3. Find Laurent series expansions of

(a) f(2) = -, at =0,0<<1.

(Answer: f(2) = "-).

(b) f(2) = - at 2 = 01.

(Answer: f(z) Σ 1

(c) f(x): (1), at 1.0-1 <1. =

(Answer: ()=(-1)"n(-1)-2

(c) f(2) = (-1) (2-3), at 1,0-1 <2.

(Answer: f(2) = 1-3 (2-1)") 2n+2 n=0 (d) f(z) = = cos 2, at z=2, 0 <-2 < ∞.

(Answer: f(z) = (-1)" 1 +Σ (-1)"2 I (2n)! (-2)2-1 (2n)! (-2)2n