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Complex variables

  1. Oct 19, 2009 #1
    1. The problem statement, all variables and given/known data

    let:
    f(z)=u(x,y)+iv(x,y)
    I want to express the following function like the one above:
    [tex]f(z)=cos(z)\equiv=\frac{1}{2}(e^{iz}+e^{-iz})[/tex]

    2. Relevant equations

    (i=sqrt(-1))
    f(z) is a complex function

    3. The attempt at a solution

    [tex]f(z)=cos(z)\equiv=\frac{1}{2}(e^{iz}+e^{-iz})[/tex]
    [tex]=\frac{1}{2}(e^{i(x+iy)}+e^{-i(x+iy)})=\frac{1}{2}(e^{x-iy}+e^{-xi+y})[/tex]

    this is where i stopped because i got stuck. any help please?
     
  2. jcsd
  3. Oct 19, 2009 #2

    Mark44

    Staff: Mentor

    Check your work in the expression before the last =.
    i(x + iy) = ix + i2y = -y + ix. The other one is OK.

    So one of your expressions will be e-y + ix = e-yeix = e-y(cos x + i sin x) = e-y cos x + i e-y sin x. Do about the same thing to the other expression and add the two together.
     
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