Definition: Euler's formula

For any real

The Euler’s formula can be seen as a definition of the complex exponential.

Trig Functions to Exponential Functions

We can derive . Adding or subtracting it with the original Euler’s formula:

Some Intuition

There are two ways to represent complex numbers:

  • Cartesian form:
  • Polar form: complex-cartesian-polar.png From , if we substitute and with and , we get

If we multiply two complex numbers, their magnitude multiply but their angle adds:

And a function with this property is the exponential function: , which is a clue that complex number can behave like exponential functions. 1 And indeed, defining works well.

Derivation

We can derive the Euler’s formula using the Maclaurin series

De Moivre’s Formula

Main: De Moivre’s formula

An application of the Euler’s formula is the De Moivre’s formula:

We can derive it with the following

Footnotes

  1. Notes on the Euler formula - Eli Bendersky’s website