To find the poles of the function \( \frac{1}{\sin(\pi z)} \) within the contour \( |z| = \pi \), we need to determine where the denominator \(\sin(\pi z)\) equals zero within this contour. 

The sine function, \(\sin(\pi z)\), has zeros at the points where \(\pi z = n\pi\), where \( n \) is any integer. This simplifies to \( z = n \).

Now, we need to determine which of these points lie within the contour \( |z| = \pi \), which means we are looking for the points \( z = n \) such that \( |n| < \pi \).

The integer values \( n \) that satisfy this condition are:
\[ -\pi < n < \pi \]

Since \(\pi\) is approximately 3.14, the integer values of \( n \) that fall within this range are:
\[ -3, -2, -1, 0, 1, 2, 3 \]

Therefore, the poles of \( \frac{1}{\sin(\pi z)} \) within the contour \( |z| = \pi \) are located at:
\[ z = -3, -2, -1, 0, 1, 2, 3 \]

Would you like more details or have any questions about this solution?

Here are 5 questions you might want to ask:

1. How do you find the residue of a pole for a given function?
2. What is the significance of the contour \( |z| = \pi \) in complex analysis?
3. How do you determine the order of a pole?
4. What are the Laurent series and how are they used in complex analysis?
5. Can you explain the concept of residue theorem?

**Tip:** When dealing with complex functions, always start by identifying singularities and checking if they lie within your specified contour.

Explore the poles of the function 1/sin(pi*z) within the contour |z|=pi. Understand how to find these poles using complex analysis techniques. This problem involves determining where the sine function equals zero within a specified contour, essential for contour integration in complex analysis.

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