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Singular Point : Pole and Residue

2. Show that the singular point of each of the following functions is a pole. Determine the order m of that pole and the corresponding residue B.
{please see attachment for functions}

Please specify the terms that you use if necessary and clearly explain each step of your solution.


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In general a pole (z0) of f(z) is of order m, if for the first whole number m the following limit exists:

lim (z-z0)^m*f(z)

Naturally if this limit exists, z0 is a pole.

The singularity is at z=0, so z0=0. Now we want to find m such that:

lim z^m*(1-cosh z)/z^3

exists. m=0:

lim (1-cosh z)/z^3 ---> lim -sinh z/(3z^2) ---> lim -cosh z/(6z) ---> infinity
z--->0 z--->0 z--->0

so the order is not 0. Now m=1:

lim z(1-cosh z)/z^3 = lim (1-cosh z)/z^2 ---> lim -sinh ...

Solution Summary

Poles and residues are investigated.