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Math Help - Domain of analyticity of a function

  1. #1
    MHF Contributor arbolis's Avatar
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    Domain of analyticity of a function

    I must find the domain of analyticity of the function f(z)=\frac{1}{\cosh z}.
    Attempt:
    Since \cosh (z)=\frac{e^{iz}+e^{-iz}}{2} is analytic over the whole complex plane, f is analytic on the whole complex plane except when \frac{e^{iz}+e^{-iz}}{2}=0.
    This is equivalent to say that if z=a+ib, a=\frac{\pi}{2}+\frac{k \pi}{2} with k\in \mathbb{Z} and \cos (2a)=-e^{b}. I've searched in google the zeros of the complex function cosh and I didn't find anything (not even in wikipedia) that showed them. Of course likely I've missed something but still. I'm out of ideas about how to go further in the expression \cos (2a)=-e^{b}. What does this represent... Any idea?
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  2. #2
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    We can write \cosh(x+yi)=\cosh(x)\cos(y)+i~\sinh(x)\sin(y).
    Now that is zero only if x=0~\&~y=\frac{(2k+1)\pi}{2}.

    Now I may have misread your question. But does this help?
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  3. #3
    MHF Contributor arbolis's Avatar
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    Yes it helped a lot. Thank you very much.
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