boundary Value Problem

How do you solve the attached boundary value problem?

Mobius/conformal Maps

(1) For j = 1,2 let R_j be the circle of diameter j/2 and center at (j/4)i. Also, let p(z) = 1/z be the inversion map. (a) If G is the region outside R_1 and inside R_2 then prove that p(G) = {z : -2 < Im z < -1}. (my work) I was able to draw the 2 circles, one inside the other, and also found the strip that it maps to ...continues

Need proof uniqueness of mobius transformations for CIRCLES

Prove: For any given circles R and R' in C_oo, there is a mobius transformation T such that T(r)=R'. Further, we can specify that T take any 3 points on R onto any 3 points of R'. If we do specify Tz_j for j=2,3,4 (distinct z_j in R), then T is unique.

Complex Integrals

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.

Use of the Cauchy Integral Formula for proofs

Cauchy Integral Formula. See attached file for full problem description.

Complex Proof

Let f be an entire function such that |f(z)| ≤ A|z| for all z in C for some fixed positive real number A. Use the attached theorem to show that f(z) = mz for some complex number m.

Derivation of Poisson Integral Formula for the Half-Plane

Derivation of Poisson Integral Formula for the Half-Plane. Attached is an outline, I just need to figure out how to justify the steps.

Complex Variables (Laurent Series, Uniform Convergence)

(1) Let G = {z : 0 < abs(z) < R} for some R > 0 and let f be analytic on the punctured disk G with Laurent Series f(z) = sum a_n*z^n (from n = -oo to oo). (a) If f_n(z) = sum a_k*z^k (from k =-oo to n), then prove that f_n converges pointwise f in C(G,C) (all continuous functions from G to C (complex)); i.e., {f_n} ...continues

Complex Metric Spaces

Let (S,d) be a metric space and define the function u(x,y) = d(x,y)/(1+d(x,y)) for all x,y in S. (a) Prove that u is a metric on S with sup u(x,y) <= 1. (my work): i did this already but want to make sure..for the triangle inequality i took a derivative of f(x) = x/1+x and went from there...also need the symmetric part ...continues

Convergent Series

How many terms of the convergent series do you need in order for the partial sum Sn to estimate the actual sum of the series correctly to four decimal places. ( The sum of the series from n=1 to infinity of (-1) ^5n divided by Ln n.)