Integrating Complex Functions in the Complex Plane

[dykuma]

Homework Statement

Evaluate the following line integrals in the complex plane by direct integration.

Homework Equations

Z= x+i y = Cos(θ) +i Sin(θ) = e^i*θ

The Attempt at a Solution

I'm not sure how to evaluated this by hand. I tried using Z= x+i y = Cos(θ) +i Sin(θ), and evaluating the integral at dθ. However, I'm not sure how to change the bounds. It seems to me that point A starts at 2pi, and then point B is at 2pi + i *(infinity). what exactly does that mean?

 

[ehild]

Homework Statement

Evaluate the following line integrals in the complex plane by direct integration.
View attachment 108889

Homework Equations

Z= x+i y = Cos(θ) +i Sin(θ) = e^i*θ

The Attempt at a Solution

I'm not sure how to evaluated this by hand. I tried using Z= x+i y = Cos(θ) +i Sin(θ), and evaluating the integral at dθ. However, I'm not sure how to change the bounds. It seems to me that point A starts at 2pi, and then point B is at 2pi + i *(infinity). what exactly does that mean?

You overcomplicate the problem. Do the integral with respect to z, as if it was a common real number. Then substitute the limits for z, using that the upper limit means z=x+iy=2pi + iy, y-->infinity.

 

[dykuma]

You overcomplicate the problem. Do the integral with respect to z, as if it was a common real number.Then substitute the limits for z, using that the upper limit means z=x+iy=2pi + iy, y-->infinity.

I see. That is what I wanted to do at first. However, I was taken back by the complex part of the upper bound. I see now that I really over complicated that problem.

Thank you!