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Thread: 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 $\displaystyle f(z)=\frac{1}{\cosh z}$.
    Attempt:
    Since $\displaystyle \cosh (z)=\frac{e^{iz}+e^{-iz}}{2}$ is analytic over the whole complex plane, f is analytic on the whole complex plane except when $\displaystyle \frac{e^{iz}+e^{-iz}}{2}=0$.
    This is equivalent to say that if $\displaystyle z=a+ib$, $\displaystyle a=\frac{\pi}{2}+\frac{k \pi}{2}$ with $\displaystyle k\in \mathbb{Z}$ and $\displaystyle \cos (2a)=-e^{b}$. 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 $\displaystyle \cos (2a)=-e^{b}$. What does this represent... Any idea?
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  2. #2
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    We can write $\displaystyle \cosh(x+yi)=\cosh(x)\cos(y)+i~\sinh(x)\sin(y)$.
    Now that is zero only if $\displaystyle x=0~\&~y=\frac{(2k+1)\pi}{2}$.

    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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