Math Problem Statement

Poles of 1/sinpiz in contour |z|=pi

Solution

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

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

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

The integer values nn that satisfy this condition are: π<n<π-\pi < n < \pi

Since π\pi is approximately 3.14, the integer values of nn that fall within this range are: 3,2,1,0,1,2,3-3, -2, -1, 0, 1, 2, 3

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

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Math Problem Analysis

Mathematical Concepts

Complex Analysis

Formulas

\( \sin(\pi z) = 0 \)

Theorems

Contour Integration

Suitable Grade Level

Advanced Undergraduate