# Complex Exponential

1. Jun 5, 2006

### Repetit

Hey!

I was wondering, is it merely a definition that

$$e^{ix}= cos(x) + i sin(x)$$

or is it actually important that it is the number e which is used as base for the exponential?

Thanks!

2. Jun 5, 2006

3. Jun 5, 2006

### HallsofIvy

Staff Emeritus
No, it's not a definition, yes, it is important the "e" be the base of the exponentials. That formula can be derived from the Taylor's series for ex, cos x, and sin x. Taylor's series depends upon the derivatives and the derivative of ex happens to be ex itself. The derivative of ax is axln(a) so corresponding formulas are more complicated.
For similar reasons, it is also important that the argument in the sine and cosine be interpreted as radian measure, not degrees.