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    Singular points and residues of 1/[(Cosh(z))-2exp(z)]

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    Classify all singular points of f(z) = 1/[(Cosh(z))-2exp(z)] and find all residues.

    © BrainMass Inc. brainmass.com March 4, 2021, 11:05 pm ad1c9bdddf
    https://brainmass.com/math/complex-analysis/singular-points-residues-392255

    Solution Preview

    We can find the singular points by equating the numerator to zero:

    Cosh(z) - 2 exp(z) = 0

    Put exp(z) = y. Then we have:

    y + y^(-1) - 4 y = 0 ---------->

    3 y^2 = 1 --------->

    y = +/- 3^(-1/2)

    We can then compute z using

    z = Log(y) + 2 pi i n

    where n is an integer and the logarithm is defined by taking the branch ...

    Solution Summary

    We explain in detail how the singularities of the function can be found, and we compute the residues at the poles.

    $2.49

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