To solve this problem, we need to analyze the singularities of the given function:

$f(z) = e^{\frac{1}{z}} + \frac{\pi(z^2 - z)}{\sin(\pi z)}$

Step 1: Identify Singularities of $f(z)$

The function $f(z)$ is composed of two terms, each of which may have singularities:

1. First Term: $e^{\frac{1}{z}}$

   • This term has a singularity at $z = 0$ due to the $\frac{1}{z}$ in the exponent. This singularity is an essential singularity at $z = 0$ because $e^{\frac{1}{z}}$ has an essential singularity at $z = 0$.

2. Second Term: $\frac{\pi(z^2 - z)}{\sin(\pi z)}$

   • The singularities of this term come from the zeros of the denominator $\sin(\pi z)$, which occurs at $z = n$ for any integer $n$ (where $n \in \mathbb{Z}$).
   • Each $z = n$ (for $n \neq 0$) is a simple pole of the second term since $\sin(\pi z)$ has a simple zero at these points.
   • At $z = 0$, we need to examine it more carefully, as it might behave differently due to the form of the numerator $\pi(z^2 - z)$.

Step 2: Analyze the Residue at $z = 0$

The problem also asks for the residue at $z = 0$ for $f(z)$.

Since $z = 0$ is an essential singularity for $e^{\frac{1}{z}}$, the residue calculation focuses on the behavior of the second term around $z = 0$. The Taylor expansion of $\sin(\pi z)$ around $z = 0$ is:

$\sin(\pi z) \approx \pi z - \frac{(\pi z)^3}{6} + \cdots$

Thus, near $z = 0$:

$\frac{\pi(z^2 - z)}{\sin(\pi z)} \approx \frac{\pi(z^2 - z)}{\pi z} = z - 1$

So, the residue of $\frac{\pi(z^2 - z)}{\sin(\pi z)}$ at $z = 0$ is $-1$. Since $e^{\frac{1}{z}}$ does not contribute a residue at $z = 0$ (due to it being an essential singularity with no residue), the residue of $f(z)$ at $z = 0$ is simply the residue from the second term, which is $-1$.

Conclusion

• Singularities of $f(z)$:
  • $z = 0$: Essential singularity.
  • $z = n$ (for $n \neq 0$ and $n \in \mathbb{Z}$): Simple poles.
• Residue at $z = 0$: $\text{Res}(f, 0) = -1$.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What is an essential singularity, and how does it differ from a pole?
2. How are residues computed for functions with essential singularities nearby?
3. Why does $e^{\frac{1}{z}}$ have no residue at $z = 0$ despite being singular?
4. How can we generalize this method to find residues at other points?
5. What is the Laurent series expansion, and how could it help in analyzing such functions?

Tip: Essential singularities are unique because functions exhibit extremely sensitive behavior around them, as captured by the Casorati-Weierstrass theorem, which states that near an essential singularity, a function can take almost any complex value.