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Math Help - Integration with branch cuts

  1. #1
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    Integration with branch cuts

    Hi,
    I find integration with branch cuts difficult to grasp.

    Here are few question I'd like to clarify.


    1.
    Take for example, w = \sqrt{z} which is mapped from z-plane to w-plane.

    In z-plane when 0\leq\arg{z}<2\pi this is mapped to only the half plane of w 0<\arg{w}<pi. This is the first branch of w
    So the z-plane has to have a cut along [0, +\infty).
    w's next branch is obtained with 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')

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

    Following procedure is used.
    \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 \sqrt{z} to be positive on top of the cut. - what is the precise meaning of this reference?
    Then he takes

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

    and by using residues
    \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.
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  2. #2
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    \sqrt{z} 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 0 < arg(\omega) < \pi, which is one of the branches.
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  3. #3
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    This is how I think equation (A) was obtained...
    \frac{\sqrt{z}}{1+z^2} = \frac{\sqrt{z}}{(z+i)(z-i)}
    choosing g_1(z) = \frac{\sqrt{z}}{z+i} and g_2(z) = \frac{\sqrt{z}}{z-i} we get
    \oint_{c1}{\frac{\frac{\sqrt{z}}{z+i}}{z-i}}dz = 2\pi i*g_1(z) and
    \oint_{c2}{\frac{\frac{\sqrt{z}}{z-i}}{z+i}}dz = 2\pi i*g_2(z). Now,
    I = \oint{\frac{\sqrt{z}}{1+z^2}dz} = 2\pi i*[g_1(z)+g_2(z)]
     = \oint_{c1}{\frac{\frac{\sqrt{z}}{z+i}}{z-i}}dz + \oint_{c2}{\frac{\frac{\sqrt{z}}{z-i}}{z+i}}dz
     = 2 \oint{\frac{\sqrt{z}}{(z+i)(z-i)}} = 2 \oint{\frac{\sqrt{z}}{1+z^2}dz} = 2I
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  4. #4
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    \sqrt{z}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.
    I have seen this sort of reference several times. 'on top of the cut' or 'on bottom of the cut'
    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.
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