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To solve problem 11 from the image, let's first extract the problem statement and then work on the solution.

### Problem Statement:
The problem asks to examine the nature of singularities of the given functions and determine whether they are poles or not. The functions provided are:

\[
\text{(a)} \quad \frac{\cos z}{z^4 - 16}
\]
\[
\text{(b)} \quad \frac{\sin z}{(z^2 + 1)(z - 3)}
\]
\[
\text{(c)} \quad \frac{1}{z^2 \sin z}
\]

### Solution:

#### (a) \(\frac{\cos z}{z^4 - 16}\):

The singularities occur where the denominator equals zero:
\[
z^4 - 16 = 0 \Rightarrow z^4 = 16 \Rightarrow z = \pm 2, \pm 2i
\]

These singularities are poles because at each singularity, the denominator becomes zero while the numerator (\(\cos z\)) is non-zero. We need to determine the order of each pole:
- Near \(z = 2\), the factor in the denominator behaves like \((z - 2)\), so it’s a simple pole.
- Similarly, for \(z = -2\), \(z = 2i\), and \(z = -2i\), each is a simple pole.

Thus, all singularities \(z = \pm 2\), \(z = \pm 2i\) are simple poles.

#### (b) \(\frac{\sin z}{(z^2 + 1)(z - 3)}\):

The singularities occur where the denominator equals zero:
\[
(z^2 + 1)(z - 3) = 0 \Rightarrow z^2 + 1 = 0 \text{ or } z - 3 = 0
\]
\[
z = \pm i, \quad z = 3
\]

These singularities are:
- \(z = \pm i\): Here, \(z^2 + 1 = 0\) gives \(z = \pm i\), which are simple poles.
- \(z = 3\): Here, \(z - 3 = 0\), which is also a simple pole.

Therefore, the singularities \(z = \pm i, z = 3\) are simple poles.

#### (c) \(\frac{1}{z^2 \sin z}\):

The singularities occur where the denominator equals zero:
\[
z^2 \sin z = 0
\]
This occurs when either \(z^2 = 0\) or \(\sin z = 0\).

1. \(z^2 = 0 \Rightarrow z = 0\): This is a pole of order 2 since \(z^2\) in the denominator contributes a double zero.
2. \(\sin z = 0 \Rightarrow z = n\pi\), where \(n\) is an integer: These are simple poles since \(\sin z\) has a simple zero at \(z = n\pi\).

Thus:
- \(z = 0\) is a pole of order 2.
- \(z = n\pi\), \(n \neq 0\) are simple poles.

### Summary:
- For function (a), the poles are at \(z = \pm 2\), \(z = \pm 2i\) (all simple poles).
- For function (b), the poles are at \(z = \pm i\), \(z = 3\) (all simple poles).
- For function (c), there is a pole of order 2 at \(z = 0\) and simple poles at \(z = n\pi\), \(n \neq 0\).

---

Would you like a more detailed explanation of any of the steps, or do you have any specific questions? Here are some related questions you might find interesting:

1. What is a singularity in a complex function?
2. How do you determine the order of a pole?
3. What is the difference between a pole and a removable singularity?
4. Can a pole be of order higher than 2?
5. How do you identify an essential singularity?
6. What happens to a function near a simple pole?
7. How can you expand a function in a Laurent series around a singularity?
8. What is the residue of a function at a pole?

**Tip:** Understanding the nature of singularities and how to classify them is crucial in complex analysis, especially when applying residue theorem for contour integration.

Explore the analysis of singularities and poles in complex functions. Detailed solutions for functions involving cosine, sine, and their implications. Suitable for advanced undergraduate students studying complex analysis.

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