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Math Help - analytic complex functions

  1. #1
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    analytic complex functions

    I am supposed to prove that f(z) = e^(-x)e^(-iy) is analytic everywhere. I will be able to do it, I think, but I do not know how to break f(z) into the real and imaginary parts (u(x.y) + iv(x,y)). Can someone please help???
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  2. #2
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    Hier.
    Attached Thumbnails Attached Thumbnails analytic complex functions-picture19.gif  
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  3. #3
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    ok, now I am supposed to find the derivative of the function. Do I just add the two partial derivatives?
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  4. #4
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    It is equal to: u_x + i*v_x
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  5. #5
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    The derivative of e^(-x)e^(-iy) equals u_x + i*v_x? Ok, I hadn't realized that.
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    Quote Originally Posted by Hollysti View Post
    The derivative of e^(-x)e^(-iy) equals u_x + i*v_x? Ok, I hadn't realized that.
    But remember to write it in standard form:
    u(x,y)+iv(x,y)

    In fact, instead we can write,
    u_y-i*v_y
    (Why?)
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  7. #7
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    Eulers formula?
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  8. #8
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    Quote Originally Posted by Hollysti View Post
    Eulers formula?
    No! The Cauchy-Riemann equations.

    If f(z) is analytic then u_x = u_y and u_y = -v_x.
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  9. #9
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    Oh, right. That makes sense. Thank you for taking time to explain that to me!
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