Complex Numbers

A complex number is a number with a real and an imaginary part, usually expressed in cartesian form

The polar form can also be expressed in terms of trigonometric functions using the Euler relationship

Euler Relationship

The trigonometric functions are related to a complex exponential by the Euler relationship

From these relationships the trig functions can be expressed in terms of the complex exponential:

This relationship is useful for expressing complex numbers in polar form, as well as many other applications.

Polar Form, Complex Numbers

The standard form of a complex number is

but this can be shown to be equivalent to the form

which is called the polar form of a complex number. The equivalence can be shown by using the Euler relationship for complex exponentials.