((sin(π*z))/(z*(z^(2)+1)^(2))) find poles

To find the poles of the function

\[
f(z) = \frac{\sin(\pi z)}{z \left( z^2 + 1 \right)^2}
\]

we need to determine the values of \( z \) where the function becomes undefined, i.e., where the denominator is zero. Poles occur when a function blows up (becomes infinite) due to a zero in the denominator while the numerator remains finite or nonzero.

### Denominator:
The denominator is:

\[
z \left( z^2 + 1 \right)^2
\]

We set the denominator equal to zero to find the poles:

\[
z \left( z^2 + 1 \right)^2 = 0
\]

#### Step 1: Solve \( z = 0 \)
The factor \( z \) gives a solution \( z = 0 \). This is a pole.

#### Step 2: Solve \( z^2 + 1 = 0 \)
The term \( z^2 + 1 = 0 \) implies:

\[
z^2 = -1
\]

\[
z = \pm i
\]

So, \( z = i \) and \( z = -i \) are additional points where the denominator is zero.

### Nature of the Poles:
- **At \( z = 0 \):** The term \( z \) in the denominator suggests that \( z = 0 \) is a simple pole.
- **At \( z = \pm i \):** Since \( (z^2 + 1) \) appears as \( (z^2 + 1)^2 \), the poles at \( z = i \) and \( z = -i \) are **poles of order 2** (since the power of the factor is 2).

### Conclusion:
The function has:
- A simple pole at \( z = 0 \),
- Poles of order 2 at \( z = i \) and \( z = -i \).

Would you like further details on how to classify these poles, or any other related concept?

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Here are 5 related questions to expand on this topic:
1. How do you determine the residue at a pole of a meromorphic function?
2. What is the significance of higher-order poles in complex analysis?
3. Can you find the Laurent series of this function around \( z = 0 \)?
4. How would the behavior of the function change near \( z = i \) or \( z = -i \)?
5. How do poles contribute to contour integrals in the residue theorem?

**Tip:** The nature of a pole (simple, double, etc.) is determined by how many times the function becomes infinite as you approach the pole.

Explore the solution to find the poles of the complex function (sin(πz))/(z*(z^2 + 1)^2). This detailed analysis covers the steps to locate and classify poles, including simple and double poles, as well as related complex analysis concepts. Ideal for undergraduate math or engineering students.

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