This seems to be a bit of a stupid question, but I can't get it right and need som help.
I've a formula that says: If $$f(z) = (z-a)^{-N}g(z)$$ then $$Res_{z=a} f(z) = \frac{g^{N-1}(a)}{(N-1)!} $$
I got a dubble pole in $z=3i$ and a function $$g(z) = \frac{e^{2iz}}{(z^2+9)^2}$$
No my problem is to apply the forumula to the function. I get that $ N = 2$ and $z=3i$ but that gives that the demoninator becomes 0 and that's not right.
Note that here, $N=4.$
$$f(z) = \frac{e^{2iz}}{(z+3i)^2}(z-3i)^{-2} $$
Now compute the $(N-1)$th derivative of $g$: $$ g'(z) = \frac{2ie^{2iz}(z+4i)}{(z+3i)^3} $$ and evaluate: $$ Res = g'(3i) = -\frac{7ie^{-6}}{108} $$