Complex Integration Along Given Path

In summary, the conversation discusses parametrizing a path on the complex plane, using Cauchy's Integral Formula and the Residue Theorem to evaluate an integral. The final answer should be a complex number, but the speaker is unsure of how to incorporate the given path into the integral. They mention a possible discrepancy between their calculated answer and the expected answer, and seek clarification on the bounds of the parameter.
  • #1
Homework Statement
I am asked to find the value of the integral I = dz / (z * (z + 4)) along the contour z = 4 * t * exp(-t * 2* pi * i) + 1, where the bounds of t are [0,1].
Relevant Equations
Cauchy's Integral Formula, Residue Theorem
From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z + 4). This gives me an answer of I = 2 * pi * i, but I know the answer is supposed to be 0.255 (from evaluating the integral directly between z=1 and z=5. Using the Residue Theorem gives me the same answer, so I am unsure of how to proceed,
 
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  • #2
I don't understand your plan. You can evaluate the integral on the closed contour, but so what? You have no idea what the integral is on the extra piece you added to the path.
 
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  • #3
Office_Shredder said:
I don't understand your plan. You can evaluate the integral on the closed contour, but so what? You have no idea what the integral is on the extra piece you added to the path.
I was parametrizing that piece as z = x & dz=dx and evaluating the integral between 1 and 5. I guess I am stuck on how to actually incorporate the given path into an integral.
 
  • #4
usersusername1 said:
From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z + 4).
What happened to the ##1/z## factor? What is the residue of ##1/(z(z+4))## at ##z=0##?
usersusername1 said:
This gives me an answer of I = 2 * pi * i,
That is for the entire closed curve, right?
usersusername1 said:
but I know the answer is supposed to be 0.255 (from evaluating the integral directly between z=1 and z=5.
Is 0.255 for the part of the closed path that you added, or is it a book answer for the original partial path? I would be surprised if the final answer was not complex.
usersusername1 said:
Using the Residue Theorem gives me the same answer, so I am unsure of how to proceed,
I have not tried to follow the details, but I think that you now have ##2 \pi i = -DesiredIntegral + 0.255##. So now you can easily find the value of ##DesiredIntegral##. (I put the minus sign in because it looks like the original path is clockwise. "the bounds of t are [0,1]" is not clear. Is it from 0 to 1 or is it from 1 to 0?)
 
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1. What is complex integration along a given path?

Complex integration along a given path is a mathematical concept in which a function is integrated over a specific path in the complex plane. It is used to calculate the area under a curve in the complex plane and has applications in various fields such as physics, engineering, and economics.

2. How is complex integration along a given path different from regular integration?

Complex integration along a given path involves integrating a function over a path in the complex plane, while regular integration is done over a real interval. Complex integration also takes into account the imaginary part of the function, making it a more complex process.

3. What is the Cauchy-Goursat theorem in relation to complex integration?

The Cauchy-Goursat theorem states that if a function is analytic in a region and its derivative is continuous in that region, then the integral of the function along any closed path in that region is equal to zero. This theorem is used to simplify complex integration problems by reducing the number of paths that need to be evaluated.

4. What are the common techniques for evaluating complex integration along a given path?

The most commonly used techniques for evaluating complex integration along a given path include the Cauchy integral formula, the residue theorem, and the method of partial fractions. These techniques involve manipulating the function and the path of integration to simplify the integral and solve for the desired result.

5. What are the applications of complex integration along a given path in real life?

Complex integration along a given path has various applications in fields such as physics, engineering, and economics. It is used to calculate the work done by a force on a moving object, the electric field around a charged particle, and the value of complex-valued financial derivatives. It is also used in signal processing and control systems to analyze and design systems with complex inputs and outputs.

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