A sinusoid can be represented as the real part of a rotating complex exponential:

$$A\cos(\omega t+\phi)=\Re\lbrace A e^{i\phi}e^{i\omega t}\rbrace.$$

The exponential

$$e^{i\omega t}$$

is a complex sinusoid with angular frequency $\omega$.

Why complex exponentials are useful

Differentiation and integration preserve the exponential form:

$$\frac{d}{dt}e^{i\omega t}=i\omega e^{i\omega t}.$$

This makes complex sinusoids natural building blocks for linear differential systems.

Frequency components

A complicated signal can often be represented as a weighted combination of sinusoids at different frequencies. Each sinusoidal component contributes an amplitude and phase.

Instead of describing only how a signal changes with time, a frequency-domain description asks how strongly different frequencies are present.

Positive and negative frequency

Euler's formula gives

$$\cos(\omega t)=\frac{e^{i\omega t}+e^{-i\omega t}}2.$$

A real sinusoid can therefore be represented using paired positive- and negative-frequency complex components.

This representation is the algebraic foundation of Fourier series and Fourier transforms.