To solve this, we aim to calculate the sum of the residues of $f(z)$ at all its finite singular points.

Given function:

$f(z) = \frac{z^2}{(z^2 + 2)^3 (z^2 - 1)^4}$

Steps:

1. Identify singularities:

The singularities of $f(z)$ occur at the roots of $(z^2 + 2)$ and $(z^2 - 1)$:

• Roots of $z^2 + 2 = 0$: $z = \pm i\sqrt{2}$
• Roots of $z^2 - 1 = 0$: $z = \pm 1$

Thus, the singular points are $z = \pm i\sqrt{2}$ and $z = \pm 1$.

2. Nature of singularities:

• $z = \pm i\sqrt{2}$: These are poles of order 3 due to the factor $(z^2 + 2)^3$.
• $z = \pm 1$: These are poles of order 4 due to the factor $(z^2 - 1)^4$.

3. Sum of residues at finite singular points:

Using the residue theorem, the sum of the residues of $f(z)$ at all finite singularities is related to the residue at infinity: $\text{Sum of residues at finite points} = -\text{Residue at infinity}.$

4. Residue at infinity:

To compute the residue at infinity, we analyze the behavior of $f(z)$ as $z \to \infty$. Rewrite $f(z)$ in terms of $w = \frac{1}{z}$ (where $w \to 0$ as $z \to \infty$):

Substitute $z = \frac{1}{w}$, so: $f(z) = f\left(\frac{1}{w}\right) = \frac{\left(\frac{1}{w}\right)^2}{\left(\left(\frac{1}{w}\right)^2 + 2\right)^3 \left(\left(\frac{1}{w}\right)^2 - 1\right)^4}.$

Simplify: $f\left(\frac{1}{w}\right) = \frac{\frac{1}{w^2}}{\left(\frac{1}{w^2} + 2\right)^3 \left(\frac{1}{w^2} - 1\right)^4}.$

Multiply numerator and denominator by $w^{18}$ (to clear all denominators): $f\left(\frac{1}{w}\right) = \frac{w^{16}}{\left(1 + 2w^2\right)^3 \left(1 - w^2\right)^4}.$

As $w \to 0$, the leading term of the numerator and denominator gives the residue at infinity. Expanding the dominant term, we observe that: $\text{Residue at infinity} = 0.$

5. Conclusion:

Since the residue at infinity is zero, the sum of the residues at all finite singular points is also zero: $\boxed{0}$

Let me know if you need further clarification or additional steps explained! Here are related questions:

1. What is the residue at $z = 1$?
2. How does the order of the pole affect the residue calculation?
3. Can the residue at infinity ever be non-zero for rational functions?
4. How can we expand $f(z)$ for large $z$ to verify the behavior at infinity?
5. Why does the residue theorem simplify calculations for rational functions?

Tip: Always check the nature of singularities before computing residues to simplify your calculations.