Hi,

I find integration with branch cuts difficult to grasp.

Here are few question I'd like to clarify.

1.

Take for example, $\displaystyle w = \sqrt{z}$ which is mapped from z-plane to w-plane.

In z-plane when $\displaystyle 0\leq\arg{z}<2\pi$ this is mapped to only the half plane of w $\displaystyle 0<\arg{w}<pi$.This is the first branch of w

So the z-plane has to have a cut along $\displaystyle [0, +\infty)$.

w's next branch is obtained with $\displaystyle 2\pi\leq\arg{z}<4\pi$ which is traversed along another surface parallel to z-plane which is joined at the positive real axis.This is the second branch of w

Is this interprtation correct?

2.

To evaluate below (in Mathews book pp-70 'Mathematical methods of physics')

$\displaystyle \int^{\infty}_{0}{\frac{\sqrt{x}}{1+x^2}dx} = I $

Following procedure is used.

$\displaystyle \oint{\frac{\sqrt{z}}{1+z^2}dx}$

On the keyhole contour similar to this

File:Keyhole contour.svg - Wikipedia, the free encyclopedia

Then he chooses $\displaystyle \sqrt{z}$ to bepositive on top of the cut.- what is the precise meaning of this reference?

Then he takes

$\displaystyle \oint{\frac{\sqrt{z}}{1+z^2}dx} = 2I$ - (A)

and by using residues

$\displaystyle \oint{\frac{\sqrt{z}}{1+z^2}dx} = \pi\sqrt{2}$ - (B)

And evaluates for I from (A) and (B)

Can u pls explain this procedure giving more elaborate explaination?

I find (A) came from thin air !

3.

Can u explain how one could do some integral and choose after selecting branch cuts.

Thank you.