Abel's Theorem in Complex analysis

I am taking Complex Analysis class and I am trying to prove Abel's theorem. a). I want to prove that if sum of a_n(z-a)^n have radius of convergence 1 and if the sum a_n converges to A then lim (r -> 1- ) of the sum (a_n r^n) = A. ( I believe z here is a complex number). b). Using Abel's theorem, prove that log2 = 1 - 1/2 ...continues

Exponential

(See attached file for full problem description with symbols and equations) --- Prove that by applying the following Corollary: Corollary. If and are analytic on a region then iff has a limit point in . ---

Potential flow theory

(See attached file for full problem description) --- The transformation (see attached) transforms a circle with unit radius... ---

Potential flow theory

(See attached file for full problem description) --- The complex potential for a flow over a body is given by... ---

Potential flow theory

(See attached file for full problem description) --- Find the resultant velocity vector induced at point A... ---

D-contour intervals

I need help on how to work out the solution to a function using the 'D-contour' (see attached file).

Complex Integral

(See attached file for full problem description)

Complex integration. Complex analysis

Ley F be entire function and suppose there is a constant M, an R > 0, and an integer n>=1 such that |f(z)| =< M|z|^n for |z| > R. Show that f is a polynomial of degree =< n. z HERE IS COMPLEX

Analytic functions/ complex integration/Complex Analysis

Let G be a region and suppose that f:G->C ( C here is complex plane)is analytic and a in G such that |f(a)|=<|f(z)| for all z in G. Show that either f(a) = 0 or f is constant.

Complex integration. Complex analysis.

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 the integral over gamma p'(z)/p(z) dz = 2 ( pi ) i n.