Line Integral and Complex Form of Green's Theorem : Compute ∫r+ (bar-z + z^2 bar-z) dz where gamma+ is a square with side = 4, centered at the origin and traced counterclockwise once

(See attached file for full problem description with equations and diagram) --- Compute ∫r+ (bar-z + z^2 bar-z) dz where gamma+ is a square with side = 4, centered at the origin and traced counterclockwise once ---

Composite Function : Continuous functions with nonvanishing first partial derivatives.

(See attached file for full problem description with equations) --- real numbers complex numbers Suppose , are continuous functions with nonvanishing first partial derivatives. Let , Show that . ---

Radius of Convergence and Abel's Theorem in Complex Analysis

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 + 1/3 - ...

Prove Exponential Identity Using an Analyticity and Congruence Corollary

Prove that e^(z+a) = (e^z)(e^a) by applying the following Corollary: Corollary. If f and g and are analytic on a region G then f ≡ g iff {z E G : f(z)=g(z)} has a limit point in G. Please see the attached file for the fully formatted problem.

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 : Find the resultant velocity vector induced at point A by the uniform stream, line source, line sink, and vortex.

Find the resultant velocity vector induced at point A by the uniform stream, line source, line sink, and vortex. Please see the attached file for the fully formatted problem. ---

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 : 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.

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