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