Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex Exponential

  1. Jun 5, 2006 #1

    I was wondering, is it merely a definition that

    [tex]e^{ix}= cos(x) + i sin(x) [/tex]

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

  2. jcsd
  3. Jun 5, 2006 #2


    User Avatar
    Homework Helper

    Have a look at http://en.wikipedia.org/wiki/Euler%27s_formula" [Broken] :smile:
    Last edited by a moderator: May 2, 2017
  4. Jun 5, 2006 #3


    User Avatar
    Science Advisor

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook