Outline of Complex Fourier Analysis

x2 + y2. Note that |z| is the distance of z from 0. Also note that |z|2 = zz. The complex exponential function e is defined to be equal to cos(θ)+ i sin(θ). Note that e is a point on the unit circle, x + y = 1. Any complex number z can be written in the form re where r = |z| and θ is an appropriately chosen angle; this is sometimes called the polar form of the complex number. Considered as a function of θ, e is a periodic function, with period 2π. More generally, if n ∈ Z, e is a function of x that has period 2π. Suppose that f : R → C is a periodic function with period 2π. The Complex Fourier Series of f is defined to be ∞