Complex Integration Along Given Path

[usersusername1]

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,

 

[Office_Shredder]

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.

 

[usersusername1]

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.

 

[FactChecker]

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 $z=0$?

This gives me an answer of I = 2 * pi * i,

That is for the entire closed curve, right?

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.

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?)