QuestionFind the remainder of the function $f(z)=\frac{e^{1 / z}}{(z-1)(z+2)^{2}}$ in the singular Point of a first-order pole.

Studdy Solution
Calculate the residue at the first-order pole $z = 1$. The residue at a simple pole $z = a$ of a function $g(z) = \frac{h(z)}{z-a}$ is given by:
$$\text{Residue} = h(a)$$
For $f(z) = \frac{e^{1/z}}{(z+2)^2}$ at $z = 1$, we have:
$$h(z) = \frac{e^{1/z}}{(z+2)^2}$$
Evaluate $h(z)$ at $z = 1$:
$$h(1) = \frac{e^{1/1}}{(1+2)^2} = \frac{e}{9}$$
Therefore, the residue at $z = 1$ is $\frac{e}{9}$.
The remainder (residue) of the function at the first-order pole $z = 1$ is $\boxed{\frac{e}{9}}$.