Math Problem Statement
Solution
To solve the given integral using Cauchy's integral formula, we start with the integral expression:
where is the contour .
Cauchy's integral formula for a function is given by:
For this problem, we will use the extended form of Cauchy's integral formula, known as the residue theorem, to solve the integral. The residue theorem states:
where are the residues of at its poles within the contour .
The function given is:
This function has poles at , , and . We need to calculate the residues at each of these poles.
Residue at :
The residue at is:
Residue at :
The residue at is:
Residue at :
The residue at is:
Sum of residues:
Now, we sum the residues:
Conclusion:
By the residue theorem, the integral is:
Therefore, the value of the integral is .
Would you like further details or have any questions?
Here are some additional questions you might have:
- How do you identify the poles of a complex function?
- Can you explain the concept of residues in complex analysis?
- What is Cauchy's integral formula, and how is it derived?
- How does the residue theorem generalize Cauchy's integral formula?
- Can you solve a different type of integral using Cauchy's integral formula?
Tip: Always verify the region of integration and ensure all poles are within the contour when applying the residue theorem.
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Math Problem Analysis
Mathematical Concepts
Complex Analysis
Residue Theorem
Cauchy's Integral Formula
Formulas
Cauchy's Integral Formula
Residue Theorem
Theorems
Residue Theorem
Suitable Grade Level
Advanced Undergraduate
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