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



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 , which we will denote by z*, is the complex number .

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

Get Principles of Fourier Analysis, 2nd Edition now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.