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Separating e^(xi) to form a-bi

  1. Feb 19, 2013 #1
    1. The problem statement, all variables and given/known data

    I am in dif eq, but just need to know how to separate a power.

    separate e^(xi) into the form a-bi, where x is a constant (in my homework, x is 4pi/3, but that's not too relevant)

    i is the imaginary number sqrt(-1)

    2. Relevant equations

    I don't know if there is some simple rule, or if I actually need to use calculus and integrals.
    The only thing I know is that e^(x+y) = e^(x)e^(y). However, I can't use that here, because the power is the multiple.

    3. The attempt at a solution

    I tried setting it equal to y = e^(xi) and taking the natural log of both sides, but it just got really messy and I ended up with a square root of i, which is not good.
     
  2. jcsd
  3. Feb 19, 2013 #2

    jbunniii

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    Are you looking for the Euler identity?
    $$e^{ix} = \cos(x) + i\sin(x)$$
     
  4. Feb 19, 2013 #3
    Are you familiar with Taylor series? As jbunniii mentioned, it is the Euler formula, but the easiest way to derive it is by using the Taylor series expansion of e^x, with x = ix, and then separate the real and imaginary terms into two series which are known to be the Taylor series for cosine and sine, respectively.
     
  5. Feb 19, 2013 #4
    ok, that makes sense, the prof did the taylor series in class......thanks
     
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