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

$z=x+iy$

where *x* and *y* are real numbers and *i* is a “complex constant” satisfying *i*^{2} = ‒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 ...

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