Math Problem Statement
Poles of 1/sinpiz in contour |z|=pi
Solution
To find the poles of the function within the contour , we need to determine where the denominator equals zero within this contour.
The sine function, , has zeros at the points where , where is any integer. This simplifies to .
Now, we need to determine which of these points lie within the contour , which means we are looking for the points such that .
The integer values that satisfy this condition are:
Since is approximately 3.14, the integer values of that fall within this range are:
Therefore, the poles of within the contour are located at:
Would you like more details or have any questions about this solution?
Here are 5 questions you might want to ask:
- How do you find the residue of a pole for a given function?
- What is the significance of the contour in complex analysis?
- How do you determine the order of a pole?
- What are the Laurent series and how are they used in complex analysis?
- 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