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Math Help - Stoke's Phenomena

  1. #1
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    Stoke's Phenomena

     \delta>0, \ \ show \ \ cosh(z) \approx \frac{1}{2}e^{z}, \ \ |arg(z)|<\frac{\pi}{2}-\delta

    and

     cosh(z)\approx \frac{1}{2}e^{-z}, \ \ \frac{\pi}{2}+\delta<arg(z)<\frac{3\pi}{2}-\delta

    not need to go along rays.
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  2. #2
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    For instance, for the first one: \cosh z=\frac{e^z}{2}(1+e^{-2z}) and |e^{-2z}|=e^{-2\Re(z)}=e^{-2|z|\cos({\rm arg}(z))} \leq e^{-2|z|\cos(\frac{\pi}{2}-\delta)}=e^{-2|z|\sin\delta} for all z such that |{\rm arg}(z)|<\frac{\pi}{2}-\delta.

    This proves that \cosh z = \frac{e^z}{2}(1+o(1)) as |z|\to\infty and |{\rm arg}(z)|<\frac{\pi}{2}-\delta, uniformly for z in this domain.
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  3. #3
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    wat

    what about the case  \delta=0
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  4. #4
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    Quote Originally Posted by pengchao1024 View Post
    what about the case  \delta=0
    Then this is false. If you choose |z|\to\infty that gets very close to the imaginary axis, \cosh z gets close to cosine of a real number, hence it can be bounded ( \cosh(ix)=\cos(x)).
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