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Math Help - Complex Analysis: find one complex number w such that w is not in the range f(z)= e^z

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    Complex Analysis: find one complex number w such that w is not in the range f(z)= e^z

    Find one complex number w such that w is not in the range of f(z)= e^z, and explain why this number is not in the range of f(z).
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    Take w=0.

    The exponential function is never 0. Why? Write x = x+iy. Then e^z = e^x(cos(y)+isin(y)). Here e^x is the usual real exponential, which is well-known never to be 0. And cosine and sine are also well-known never to be 0 simultaneously, hence e^z is non-zero for all z.
    Last edited by HappyJoe; October 3rd 2010 at 04:08 AM. Reason: Missed two words.
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