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
Example: Use the Euler's formula to compute
Some Intuition
There are two ways to represent complex numbers:
- Cartesian form:
- Polar form:
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:
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
Related
- For
where is a complex number given by , we call as the complex frequency