Complex Exponential Form

Euler's Formula

Definition

For , where the right side is a complex number in polar form

We can sorta prove this with a Taylor Series:

Example - Euler's Formula

Euler and complex numbers:


Euler noticed that this follows the same pattern as the derivatives of sine/cosine (i.e as you take more derivatives, it "cycles" through).


which is Euler's Formula!

Note that we got the values for and from the two examples above.

Complex Exponential Form

Definition

Since the complex exponential form is not unique,

Complex Exponential Form Multiplication

Note

Taking gives , which is consistent with the rules of multiplication of exponential functions

Complex Exponential Form Powers

Note

For

Taking gives , which is consistent with the rules of powers of exponential functions

Euler's Identity

Identity

Consider Euler's Formula with , . It follows that