Complex Analysis: Line Integrals

[tarheelborn]

Homework Statement

I have a problem as follows:
Let $$\gamma=\beta+[e^2\pi,1]$$ where $$\beta$$ is given by $$\beta(t)=e^{t+it}$$ for $$0\leq 2 \leq \pi$$. Evaluate $$\int_\gamma z^{-1} dz$$ .

Homework Equations

The Attempt at a Solution

I know that I need to parameterize the path and I have come up with the path as $$[e^2\pi,1]$$. I also know that I need to integrate this problem along the path and then integrate the function and add the parts together. However, I can't seem to come up with the right way to accomplish this task. I will appreciate your help. Thank you.

 

[mighty2000]

Hello,

Thank you for reaching out for help with your problem. It seems that you have already made some good progress in understanding what is required for this problem. As you mentioned, the first step is to parameterize the path \gamma. This means that you need to express \gamma in terms of a single variable, such as t, so that you can integrate along the path.

In this case, the given \gamma can be parameterized as \gamma(t) = e^{t+it} for 0\leq t \leq 2\pi. This is because \beta(t) is already given as e^{t+it} for 0\leq t \leq 2\pi, so you can simply add the constant [e^2\pi,1] to it to get the full path \gamma.

Next, you need to substitute this parameterized path into the integral \int_\gamma z^{-1} dz. This will give you an integral in terms of t that you can solve using standard integration techniques. Once you have the result, you can simply plug in the limits of integration, 0 and 2\pi, to get your final answer.

I hope this helps guide you in the right direction. Let me know if you have any further questions or need more clarification. Good luck with your problem!