Principles of Fourier Analysis, 2nd Edition

6

Elementary Complex Analysis

Fourier analysis could be done without complex-valued functions, but it would be very, very awkward.

6.1    Complex Numbers

Recall that z is a complex number if and only if it can be written as

$z=x+iy$

where x and y are real numbers and i is a “complex constant” satisfying i² = ‒1. The real part of z, denoted by Re[z], is the real number x, while the imaginary part of z, denoted by Im[z], is the real number y. If Im[z] = 0 (equivalently, z = Re[z]), then z is said to be real. Conversely, if Re[z] = 0 (equivalently, z = i Im[z]), then z is said to be imaginary.

The complex conjugate of $z=x-iy$, which we will denote by z*, is the complex number $z^{*}=x-iy$.

In the future, given any statement like “the complex number ...