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.

See attached file...it is a full induction proof of CauchyIntegralFormula, with the base case step missing. All I have to do is show that it holds for "n=1", using the rest of the proof as an example...however i am having trouble showing it.

Given: Integral from zero to infinity of cos(x^2) dx = integral from zero to infinity of sin(x^2) dx = 1/2 * (square root of pi/2).
These can be evaluated by considering cos(x^2) = Re(e^ix^2) and sin(x^2) = Im(e^ix^2).
1.) Integrate the function f(z) = e^i(z)^2 around the positively oriented boundary of the sector 0 <= r

Three questions to do with Cauchy's integral formula are evaluated in clear, easy to understand steps.
1. Show that
see attached
where C is the semicircule |x|=R>1, lm(x)>=0 from x=-R to z=R.
2. Use Cauchy's integralformula for derivatives to evaluate
see attached
where |z-2|=2 is oriented clockwise.
3. Evaluate

Use the given information:
the functions g:[a,b]->R and h:[a,b]->R are continuous with h(x) >= 0 for all x in [a,b], andthere is a point c in (a,b) such that:
theintegral from a to b of h(x)g(x)dx =
g(c) times theintegral from a to b of h(x)dx.
to show that theCauchyIntegral Remainder Theorem implies the Lagrang

(See attached file for full problem description and embedded formulae)
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Why can (1) be regarded as a special case of (2)?
(1) Cauchy's Integralformula (no need to prove):
is a simple closed positively oriented contour.
If is analytic in some simply connected domain D containing
and if is any point inside ,

Complex Differentiation
1) Suppose that an analytic function f defined on the whole of C satisfies Re(f(z))=0 for all z in C. Show that f is constant.
2i) Verify that u=x2-y2-y is harmonic in the whole complex plane.
2ii) Suppose f(x,y)=u(x,y)+iv(x,y). TheCauchy-Riemann equation state that: ux=vy and uy=-vx. For u=x2-

(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@

Give an example of each of the following or argue that such a request is impossible:
1) A Cauchy sequence that is not monotone.
2) A monotone sequence that is not Cauchy.
3) A Cauchy sequence with a divergent subsequence.
4) An unbounded sequence containing a subsequence that is Cauchy.