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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.

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Solution Summary

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

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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 ...

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