Complex Integration Homework: Answers and Guidance

In summary, the conversation is about using the Cauchy Integral Formula to solve an integral problem involving singularities. The speaker explains their method of breaking the integrand into partial fractions and using the formula for each integral. They also mention a question about integrating "around each singularity separately" and eventually come to a solution by breaking the integral into two smaller contours and applying the Cauchy Integral Formula to each one. The result is the exponential form of sin(t).
  • #1
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Homework Statement



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Homework Equations





The Attempt at a Solution



I did (i) by breaking the integrand into partial fractions and then using the Cauchy Integral Formula for each integral. I got the correct answer.

What does (ii) even mean? WHat does it mean to integrate "around each singularity separately"? The singularities are i and -i but I figure the only way to do this is partial fractions. Any help would be great.
 

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  • #2
Alright, I got it. In case anyone ever comes across this:

All I did was break it down into two smaller contours, each one containing one of the singularities but not the other. Then the integral over the entire C is equal to the sum of two integrals; one over C1 (containing one singularity) and the other over C2 (containing the other singularity). Each integrand is the same, but you're just taking one integral over C1, plus the other integral over C2. Then apply the Cauchy integral formula to each integral and you end up with the exponential form of sin(t).
 

1. What is complex integration?

Complex integration is a mathematical technique used to calculate the integral of a complex-valued function over a complex domain. It is an extension of the traditional integral calculus, which deals with real-valued functions over real domains.

2. Why do we need complex integration?

Complex integration is important in many areas of mathematics, including complex analysis, differential equations, and physics. It allows us to solve problems that cannot be solved using traditional real-valued integration techniques.

3. How do I solve complex integration problems?

There are several methods for solving complex integration problems, including the contour integration method, the Cauchy integral theorem, and the Cauchy integral formula. It is important to understand the properties of complex numbers and how they behave under integration.

4. What are some common mistakes when solving complex integration problems?

One common mistake is not properly understanding the properties of complex numbers and their behavior under integration. Another mistake is not choosing the correct method for solving the problem. It is also important to be careful with algebraic manipulations and to check for convergence of the integral.

5. Are there any tips for approaching complex integration problems?

Some tips for approaching complex integration problems include understanding the geometric interpretation of complex numbers, choosing the appropriate integration method, and being careful with algebraic manipulations. It is also helpful to practice solving a variety of problems to become familiar with different techniques.

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