I'm trying to find the residues of the functions at the points:
1) $f(z) = \dfrac{e^z}{z^2(1-2z)^2}$ at $z = \dfrac{1}{2}$.
So, we have a simple pole and can I take the derivative of the denominator? But why does this seem to still fail?
2) $f(z) = \dfrac{e^2z}{6 \cosh(z) - 10}$ at $z = \log 3$.
I am not sure why we have a pole at $z = \log 3$; it seems like from the Taylor expansion of $\cosh(z)$, nothing happens here.
3) $f(z) = (z^4 + z^2+1)\sin\left(\dfrac{1}{z}\right)$ at $z = 0$.
I think of the residue at infinity where I say: $\frac{1}{z^2}f(\frac{1}{z})$, but not sure if this would apply.