Complex line integral over x + y = 1

[Verdict]

Homework Statement

Homework Equations

I can't think of many to begin with. I've mainly been dealing with the simple forms of Cauchy's theorem so far, such as the Cauchy-Goursat theorem, and also Cauchy's integral formulas. However, these don't seem to have any direct implications here.

The Attempt at a Solution

Alright, so I first used partial fraction expansion to rewrite the integral as
i/4 [integral over dz/(z+2i) - integral over dz/(z-2i)]
However, then I have to parametrize the curve. Now, I tried using z = x + iy, and then parametrising the curve so that x = 1 - y, but that doesn't really seem to be going anywhere. I also don't see how that relates to the hint, nor do I see a way to start solving the problem from the hint.

Could someone maybe provide me with a tip as to what I should be doing?

Thanks in advance,
Verdict

 

[Dick]

Homework Statement

Homework Equations

I can't think of many to begin with. I've mainly been dealing with the simple forms of Cauchy's theorem so far, such as the Cauchy-Goursat theorem, and also Cauchy's integral formulas. However, these don't seem to have any direct implications here.

The Attempt at a Solution

Alright, so I first used partial fraction expansion to rewrite the integral as
i/4 [integral over dz/(z+2i) - integral over dz/(z-2i)]
However, then I have to parametrize the curve. Now, I tried using z = x + iy, and then parametrising the curve so that x = 1 - y, but that doesn't really seem to be going anywhere. I also don't see how that relates to the hint, nor do I see a way to start solving the problem from the hint.

Could someone maybe provide me with a tip as to what I should be doing?

Thanks in advance,
Verdict

Think about a circle with large R centered on the origin. It will intersect your line C in two points. Now imagine a new contour that follows your line between the two intersection points, then follows about half the circle in a counterclockwise direction until it intersects the line again. Can you say what the integral is over that curve using a Cauchy theorem? Now imagine R goes to infinity. What can you say about the contribution of the circular part of the contour?