Complex integrals

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3) (2) Evaluate the integral over & of dz/ z^2 -1 where & is the circle |z-i|=2 (3)Evaluate the integral over & of (z-1/z) dz where & is the line path from 1 to i (4) Evaluate the integral between 2pi and 0 of e^-i@ . e ^e^i@ d@ where @=theta (5) Show that the integral over |z|=2 of ( z/z-1)^n = 2pi ni

Complex function: Decomposition into its real and imaginary part.

Given a function w = f(z) = z^2 then decompose f(z) into its real and imaginary parts.

Find the existence of a limit in a complex value function

Find the existence of a limit in a complex value function. Please see the attached file.

Harmonic complex conjugates

Given u = y^3 - 3x^2y, find f(z) = u + iv such that f(z) is analytic. The solution is detailed and well presented.

Complex Numbers: Logarithms

Given: z1 = i and z2 = -1-i Show that: Log (z1 z2) = Log z1 + Log z2