Let C be the boundary of the square of side length 4, centered at the origin, with sides parallel to the coordinate axes, and traversed counterclockwise. Evaluate each of the attached integrals.

Contours and the Cauchy Integral Formula are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3)
(2) Evaluate theintegral over & of dz/ z^2 -1 where & is the circle |z-i|=2
(3)Evaluate theintegral over & of (z-1/z) dz where & is the line path from 1 to i
(4) Evaluate theintegral between 2pi and 0 of
e^-i@ . e ^e^i@ d@

I want to check my answer:
Evaluate the following integrals:
integral over gamma for (sin z)/z dz, given that gamma(t) = e^(it) , 0=the exponential function)
My work:
sin z = z - z^3/3! + z^5/5! + ... + (-1)^n (z^(2n-1))/(2n-1)! + ...
divide by z we get
(sin z)/z = 1 - z^2/3! + z^4/5!

Use Cauchy's formula for the derivative to prove that if f is entire and
|f(z)|≤ A|z|² + B|z| + C for all zεC,
then f(z) = az² +bz + c
Please see attached for full question.

29.18
Let f be a differentiable on R with a = sup {|f ′(x)|: x in R} < 1.
Select s0 in R and define sn = f (sn-1) for n ≥ 1. Thus s1 = f (s0), s2 = f(s1), etc
Prove that (sn) is a convergence sequence. Hint: To show (sn) is Cauchy, first show that |sn+1 - sn| ≤ aּ|sn - sn-1| for n ≥ 1.

Let P(z) be polynomial of degree n and let R>0 be sufficiently large so that p never vanishes in { z: |z| >= R}. If gamma(t) = Re^(it), 0 =< t =< 2 pi, show that theintegral over gamma p'(z)/p(z) dz = 2 ( pi ) i n.

Calculate the following integral...
Please see attached for full question.
Solution. Consider a close contour C shown above, where C consists of and a line segment from -R and R. Consider positive orientation, namely, clockwise. Choose r large enough so that are in the region covered by C.
Let . By residual Theorem

Hello. Thanks for help! I will use * to indicate a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. This is the probelm:
Use Riemann's method to solve theCauchy problem:
u*xx + 4u*xy +3u*yy = 1, u=1 and u*n = square root of 5 times x, on the intial
curve y=2x.
If this