I am trying to find the residue of the function $$f(z)=(z+1)^2e^{3/z^2}$$ at $z=0$. I tried the following in Mathematica
Residue[(z+1)^2*Exp[3/z^2],{z,0}]
which remains unevaluated. Computing this by hand gives the value of $6$. What is going on?
I’ve noticed that Mathematica has a problem with the Laurent series of $e^{3/z^2}$ at $z=0$.
You could use SeriesCoefficient instead:
SeriesCoefficient[(z+1)^2 Exp[3/z^2], {z, 0, -1}]
6
Addendum
Another possibility is to note that the residue at 0 and the residue at infinity must sum to zero, since they are the only singularities of the function. Hence we can do:
- Residue[(z + 1)^2 Exp[3/z^2], {z, Infinity}]
6
which is the same answer as before.
Or integrate around zero
Integrate[(z + 1)^2 Exp[3/z^2], {z, 1, I, -1, -I,
1}]/(2 Pi I) // Simplify
(* 6 *)
Residue: "Residues are not defined at branch points". Isn't $0$ a branch point for your function? – MarcoB Apr 30 '18 at 21:31