Principal Branch and Complex Powers

How do you find (and what is it) the principal branch of the following complex powers: (e/2)(1-i(sqrt(3)))^(3pi*i) and (1-i)^(4i)

Partial Differential Equations : Boundary Value Problem

Please see the attached file for the fully formatted problems.

Mobius Transformations and 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}. (b) Prove that e^(pie*z) maps the strip {z : -2 < Im z < -1}onto the upper half-plane H_u. (c) Use the pr ...continues

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.

Contours and the Cauchy Integral Formula

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

Use the Cauchy Integral Formula to show that where the unit circle is oriented counterclockwise, and use this fact to show that Please see the attached file for the fully formatted problems.

Complex Proof : Analytic Functions

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

If is analytic in a domain containing the x-axis and the upper half-plane and in this domain, then the values of the harmonic function in the upper half-plane are given in terms of its values on the x-axis by (y>0). Below is an outline for the derivation, I just need to figure out how to justify the steps. ...continues

Complex Variables, Laurent Series and 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. (b) If S = C (complex) and d is the usual Euclidean metric d(z,w) = abs(z-w), then prove that sup u(z,w) = 1. (c) For 0 < r < 1, show that u(x,y) < r if and only ...continues