Please specify the terms that you use if necessary and clearly explain each step of your solution.
In general a pole (z0) of f(z) is of order m, if for the first whole number m the following limit exists:
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
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 ...
Poles and residues are investigated.