I'm having a textbook at hand (Complex Analysis by Sakarchi, Stein) and I don't understand something.

At page 78, he wants to prove that

.

He says that

has 2 simple poles, one at

and the other at

. Until now everything's fine.

Now in order to calculate the integral, he chose a contour

where this curve is the half circle with radius R, centered at the origin in the upper half plane.

He then says that the residue of

at 1 is

, I still follow him.

Now he says that

. That's fine since the pole at 1 is enclosed into the curve

.

**Now this is where I don't understand. **
He goes on to say "let C be the large half circle of radius R. We see that

. The integral goes to

as

. Therefore in the limit we find

".

------------------------------------------------------

1)He never defined B and M (at least in the previous pages). I guess that they are arbitrary.

2)I don't see the necessity of finding a upper bound of the integral. (Why did he did so?)

3)He never found a curve enclosing the other pole! So how can the integral whose limits are

and

be calculated via the residue theorem for a single pole while the function has 2 poles? Is it because Sakarchi's curve encloses all real numbers when R tends to

?

If so then I could simply chose a curve enclosing all real numbers when I take the limit, but not enclosing any poles, and the contour integral would be worth

according to Cauchy's integral theorem. Which makes no sense since the integral is worth

.

Thanks for reading and helping.