Complex Exponential: Why is e Used?

[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!

 

[TD]

Have a look at http://en.wikipedia.org/wiki/Euler%27s_formula"

 

[HallsofIvy]

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 e^x, cos x, and sin x. Taylor's series depends upon the derivatives and the derivative of e^x happens to be e^x itself. The derivative of a^x is a^xln(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.