Integrals : Contours and Paths

By finding an antiderivative, evaluate each of these integrals, where the path is any contour between the indicated limits of integration. (See attachment for full question)

Integral of a Principal Branch

Show that the integration from -1 to 1 z^i dz = ((1+e^-pi)/2)*(1-i)where zi denotes the principal branch... (See attachment for full question)

Complex Variable - 500 Level Undergraduate Class

Apply the Cauchy-Goursat theorem to show that... (See attachment for full question)

Cauchy-Goursat Theorem : Proving an Integral Equality

Please see the attached file for the fully formatted problems.

The problems are from complex variable class, 500 level in undergraduate.

The problems are from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution. If there is anything unclear in the problem, please tell me. Thank you very much.

Contours and Analytic Functions

Let f denote a function that is continuous on a simple closed contour C. Prove that the function...is analytic at each point z interior to Cand that....at such a point. Please see the attached file for the fully formatted problem.

Analytic Functions and Closed Contours

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Cachy's Inequality

Let f be an entire function such that |f(z)|≤A|z| for all z, where A is a fixed positive number. Sow that f(z) = az where a is a complex constant. Please see the attached file for the fully formatted problem.

Minimum of Closed, Continuous Analytic Function

4. Let a function f be continuous in a closed bounded region R, and let it be analytic and not constant throughout the interior of R. Assuming that f(z) does not equal 0 anywhere in R, prove that f(z)f has a minimum value n in R which occurs on the boundary of R and never in the interior. Do this by applying the corresponding re ...continues

Minimum Value of Closed, Continuous Analytic Function

5. Use the function f(z) = z to show that in Exercise 4 the condition f(z) does not equal 0 anywhere in P is necessary in order to obtain the result of that exercise. That is, show that |f(z)| can reach its minimum value at an interior point when that minimum value is zero. Please see the attached file for Exercise 4 and the ...continues