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## Homework Statement

Hi all, could someone help me run through my work for these 2 integrals and see if I'm in the right direction? I'm feeling rather unsure of my work.

1) Evaluate ##\oint _\Gamma Z^*dz## along an anticlockwise circle of radius R centered at z = 0

2) Calculate the contour integral ##\int _C z^n dz## where n ∈ N, and C is a semi-circle contour as shown:

## Homework Equations

## The Attempt at a Solution

1)

The contour is parameterized by ##Z = Re^{i\theta}##, with ##0 <\theta < 2\pi##. Therefore:

##\oint _\Gamma Z^*dz = \int_{\theta _ 1}^{\theta _2} d\theta \frac{dz}{d\theta} f(z(\theta))## and,

##\int_{\theta _ 1}^{\theta _2} d\theta ( Rie^{i\theta}) (Re^{-i\theta})## thus giving

##R^2i\int_{\theta _ 1}^{\theta _2} d\theta = 2\pi R^2 i##

2)

I define another integral ##I' = \oint_\Gamma Z^n dz## where ##\Gamma## forms a closed loop of the semi-circle with original contour C and an additional contour from 0i to -i that I define as C'

I then have,

##I' = \oint_\Gamma Z^n dz = \int_{C'} Z^n dz + \int_{C}Z^n dz ##

According to Cauchy's integral theorem, ##I'## on the whole should give 0 as ##Z^n## is analytic everywhere in ##\Gamma##, leaving me with

##- \int_{C'} Z^n dz = \int_{C}Z^n dz ##

C' is parameterized by ##Z = it## with ##t## ranging from 0 to -1, hence

##- \int_{C'} Z^n dz = \int_{0}^{-1} dt \frac{dz}{dt} f(z(t))##

##- \int_{C'} Z^n dz = \int_{0}^{-1} dt(i)(it^n)##

## \int_{0}^{-1} dt(i)(it^n) = i^{n+1} \int_{0}^{-1} t^n dt = (i^{n+1})[\frac{-1^{n+1}}{n+1}]##

## \int_{C}Z^n dz = - \int_{C'} Z^n dz = (-1) (i^{n+1})[\frac{-1^{n+1}}{n+1}] = (i^{n+1})[\frac{-1^{n+2}}{n+1}]##

this then gives 2 solutions for odd and even ##n##.

## \int_{C}Z^n dz = \frac{-i^{n+1}}{n+1}## for odd ##n##

## \int_{C}Z^n dz = \frac{i^{n+1}}{n+1}## for even ##n##

Help is greatly appreciated!