Proving the fundamental theorem of algebra by using Rouche's theorem.

How do I prove the fundamental theorem of algebra by using Rouche's theorem?

Finding the residue of an expression.

Find Residue{e^(2z)/(z-1)^4, z=1}

Functions:Finding a residue.

Find the residue of F(z)=cot(z)coth(z)/z^3 at z=0

Functions: Finding the residue

Find the residue of f(z)=z^7e^(1/z) at z=0

Proving that an equation is harmonic.

Prove that U(x,y)=e^(-x)[xsiny-ycosy] is harmonic.

Harmonic Function

Prove that the function U(x,y)=2x(1-y) is harmonic

Moivre-Laplace Formula

Moivre-Laplace formula exp(ix) = cos(x) + i sin(x), where i = (-1)^(1/2) , and which is widely used in different items of mathematics is usually deduced from the Maclaurin expansions of the functions involved. But the theory of Taylor (Maclaurin) expansions is a part of more general theory developed in the course of the fun ...continues

Two ways to prove a theorem in complex numbers theory.

Let z be a complex number z=x+iy x <>0 and y<>0 Prove: 1. If z+1/z is real then |z|=1 2. If |z|=1 then z+1/z is real

Evaluate an improper integral involving trig functions using Jordan's Lemma.

Use residues to evaluate this improper integral Int(from 0 to inf)[cos(ax)/(x^2+1)]dx (a>0) (See attachment for better description.)

Isolated Singularities

Discuss isolated singularities.