This refers to the upper half of the w plane , which is one of the branches.to be positive on top of the cut. - what is the precise meaning of this reference?
Hi,
I find integration with branch cuts difficult to grasp.
Here are few question I'd like to clarify.
1.
Take for example, which is mapped from z-plane to w-plane.
In z-plane when this is mapped to only the half plane of w . This is the first branch of w
So the z-plane has to have a cut along .
w's next branch is obtained with 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')
Following procedure is used.
On the keyhole contour similar to this
File:Keyhole contour.svg - Wikipedia, the free encyclopedia
Then he chooses to be positive on top of the cut. - what is the precise meaning of this reference?
Then he takes
- (A)
and by using residues
- (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.
I have seen this sort of reference several times. 'on top of the cut' or 'on bottom of the cut'to be positive on top of the cut. - what is the precise meaning of this reference?
This refers to the upper half of the w plane , which is one of the branches.
Isn't the cut in z plane? so how would you say it is the upper half of the w plane?
There can be several surfaces created on a branch cut which can be mapped to any part of the w plane. So when it is said on top of the branch cut is it unabiguos?
thanks.