Finding the Assembly for Two Paths: Step-by-Step Guide

[asi123]

Homework Statement

Hey guys.
I have this two paths as you can see in the picture and I need to find their assembly (I hope I said it correctly).
Which one is correct, the right or the left?

Thanks.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Well, "assembly" isn't the correct English. "Union" of the two sets or "combination" of the two paths woud be better.

In any case, the problem, as I interpret this is to integrate some function from -R to R along the real line, then integrate from R to -R along the upper half circle with radius R. On the left, \gamma1 seems to be the line y= x or, in terms of complex numbers, t+ it, for t from -R to R. No, that is not at all what is given. But the picture on the right is not clear. You seem to be indicating that \gamma1 is raised up to some non-zero y, or in terms of complex numbers, t+ ai for some positive a. That is also not correct. \gamma1 is given as t+0i, not t+ some non-zero number times i. You should be showing \gamma1 running on the real axis, not above it.

 

[asi123]

Well, "assembly" isn't the correct English. "Union" of the two sets or "combination" of the two paths woud be better.

In any case, the problem, as I interpret this is to integrate some function from -R to R along the real line, then integrate from R to -R along the upper half circle with radius R. On the left, \gamma1 seems to be the line y= x or, in terms of complex numbers, t+ it, for t from -R to R. No, that is not at all what is given. But the picture on the right is not clear. You seem to be indicating that \gamma1 is raised up to some non-zero y, or in terms of complex numbers, t+ ai for some positive a. That is also not correct. \gamma1 is given as t+0i, not t+ some non-zero number times i. You should be showing \gamma1 running on the real axis, not above it.

Got you
So it actually the half circle over there together with the diameter, this is my path.

Thanks a lot and also thank you for the English correction

 

[asi123]

Well, the first part of the question ask me to find the integral in red (in the pic).
Is it right what I did?

 

[HallsofIvy]

What you have is correct but this looks more like a problem where you are expected to evaluate the integral around the closed path by using Residues. The integrand has poles of order 1 at i, -i, 2i, and -2i, of which i and 2i are inside the closed path.

 

[asi123]

What you have is correct but this looks more like a problem where you are expected to evaluate the integral around the closed path by using Residues. The integrand has poles of order 1 at i, -i, 2i, and -2i, of which i and 2i are inside the closed path.

Oh, yeah, you right, much easier.
And the points that are outside of the closed path equals to 0, right?

Thanks a lot.