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 functions of complex variable. As the Moivre-Laplace formula has numerous applications outside this theory, it seems reasonable to deduce it without references to Maclaurin series. Problem. To prove the Moivre-Laplace formula: exp(ix) = cos(x) + i sin(x) without use of the Maclaurin expansions.

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

Isolated Singularities