Show that transformation W (Z) = (a Z + b) / (c Z + d) of the upper half of a complex plane is 1-1 and onto the upper half plane if a, b, c, and d are real and satisfy condition a d > b c

Show that transformation W (Z) = (a Z + b) / (c Z + d) of the upper half of a complex plane is 1-1 and onto the upper half plane if a, b, c, and d are real and satisfy condition a d > b c

Continuity complex plane

Let G be an open subset of C ( complex plane) and let P be a polygon in G from a to b. Use the following 2 theorems to show that there is a polygon Q in G from a to b which is composed of line segments which are parallel to either the real or imaginary axes. The 2 theorems are: 1). Theorem: Suppose f: X --> omega is continuous and X is compact; then f is uniformly continuous. ( of course we are talking about complex plane remember that) 2).Theorem: If A and B are non-empty disjoint sets in X with B closed and A compact then d(A,B) > 0. Please I want a very detailed answer and justify every claim or statement in the solution. Please show where each theorem was used and why..I w... click for more

Find the radius of convergence for each of the following power series

1). Find the radius of convergence for each of the following power series. Please check my solution for this problem: a). sum ( n = 0 to infinity) a^n z^n, a is a complex number. My solution: R( radius of convergence) = lim |a_n/a_n+1) = lim | a^n/a^(n+1)| = 1/|a| b). Sum ( n=0 to infinity) = lim|a^(n^2)*z^n, a is complex number. My solution: R = lim|a^(n^2)/a^(n+1)^2| = 1/|a^(2n+1)| c). sum ( n= 0 to infinity) k^nz^n, k is an integer, k doesn't equal to 0. My solution: R= lim|k^n/k^(n+1)| = 1/|k| d). sum ( n=0 to infinity) z^n! My solution: R = lim|1| = 1. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2).Now I am stuck with this problem...I want a ... click for more

Analytic functions

1) Show that the real part of the function z^(1/2) is always positive. 2) Suppose f: G --> C ( C complex plane) is analytic and that G is connected. Show that if f(z) is real for all z in G, then f is a constant.

Analytic functions in complex plane

1). Determine the set A such that For r > 0 let A ={w, w = exp (1/z) where 0<|z|