(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In each of the following problems, evaluate the integral of the function over the given path. All paths are positively oriented (counterclockwise). In some cases Cauchy's theorem applies, and in some it does not.

2. Relevant equations

Cauchy's Theorem

Let [tex]f[/tex] be differentiable on a simply connected domain G. Let [tex]\Gamma[/tex] be a closed path in G. Then

[tex]\int_{\Gamma} f(z)dz = 0[/tex]

3. The attempt at a solution

Alright, let me see if I can explain this clearly. For the problem I am stuck on, [tex]f(z) = z^2 + Im(z)[/tex] and [tex]\Gamma[/tex] is the square with verticies [tex]0,-2i,2-2i,2[/tex]

So this is basically a square in the lower right quadrant (positive real, negative imaginary) with the upper left point being the origin. Now, I had a different problem in which [tex]\Gamma[/tex] was given to be [tex]|z|= 2[/tex], and the function, [tex]f(z) = Re(z)[/tex], and so I was able to use Euler's formula to put the function in terms of [tex]cos(t) +isin(t)[/tex], and [tex]Re(z) = cos(t)[/tex]. Taking the derivative of z(t) is easy, so I was able to evaluate the integral and get the correct answer. Now, I didn't know why Cauchy's formula didn't apply in that problem, as isn't f(z) differentiable on and within the domain enclosed by [tex]\Gamma[/tex] ? This is where I was confused at first.

But back to the current problem, [tex]\Gamma[/tex] is not the equation for a circle, so I do not know how I am supposed to interpret Im(z), or how to determine if it is differentiable on and within the domain enclosed by [tex]\Gamma[/tex] (the square).

Can anyone clarify or help me understand how to parametrize the equation, and to help me determine if Cauchy's equation applies here (i.e. whether the answer is zero, or whether I have to perform the integration)? Thanks for your help.

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# Homework Help: Cauchy's Integral Theorem Problem

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