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Math Help - Hyperbolic function equations

  1. #1
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    Hyperbolic function equations

    The question:
    Find all solutions z \in C of sinh(z) = 0

    My attempt:
    sinh(z) = sinh(x)cos(y) + icosh(x)sin(y)

    Equating real and imaginary parts:
    sinh(x)cos(y) = 0 (1)
    cosh(x)sin(y) = 0 (2)

    For (1):
    sinh(x) = 0 when x = 0, and cos(y) = 0 when y = (\frac{1}{2} + k)\pi; k \in Z

    Sub these values into (2):
    cosh(0) = 1 (therefore, we need sin(y) = 0)
    sin((\frac{1}{2} + k)\pi) is never 0.

    So it appears that it never equals zero. However, my text states that it does when z = ik\pi Where have I gone wrong? Thanks.
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  2. #2
    MHF Contributor chisigma's Avatar
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    Setting z= i x is sinh z = i sin x so that the function vanishes for x= k \pi -> z= i k \pi...

    Kind regards

    \chi \sigma
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