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The image shows a mathematical question related to calculating the residue of a function at a specific point. The question asks to find the residue of the function \( \frac{9z+1}{z^2(z+1)} \) at \( z = i \). The options provided are:

- (a) \( \pi i \)
- (b) \( -\pi i \)
- (c) \( 5i \)
- (d) \( -5i \)

### Solving the Residue

To solve this, we need to compute the residue of the function at \( z = i \). Let's break it down:

1. The given function is \( f(z) = \frac{9z+1}{z^2(z+1)} \).
2. To compute the residue at \( z = i \), first check whether there is a pole at \( z = i \). Since \( z = i \) does not make the denominator zero (because neither \( z^2 \) nor \( z+1 \) vanish at \( z = i \)), we conclude there is no pole at \( z = i \), so the residue is 0.

However, the options seem to indicate the residue is not zero, suggesting the problem might involve calculating the residue at a pole or singularity of another kind. Given the mismatch, it would be important to verify or reconsider the context.

Would you like me to revisit this or dive deeper into a particular aspect of the residue calculation?

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Here are 5 questions to expand on this:

1. How do you calculate residues for poles of different orders?
2. What role does the Laurent series play in finding residues?
3. Can the residue theorem be applied to evaluate integrals?
4. How do you handle removable singularities when computing residues?
5. What are common mistakes to avoid when finding residues?

**Tip:** When working with residues, always ensure that you correctly identify the type of singularity (simple, removable, or higher-order) at the point of interest.

Find the residue of \( \frac{9z+1}{z^2(z+1)} \) at \( z = i \).

This problem explores the calculation of the residue of the function f(z) = (9z + 1) / (z^2(z + 1)) at z = i. It covers key concepts in complex analysis, including singularities, poles, and the application of the residue theorem.

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