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