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Math Help - I can't interpret this branch cut choice....

  1. #1
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    I can't interpret this branch cut choice....

    Hi,

    I'm reading a PhD dissertation. The author mentioned a branch cut when calculating the multivalued function k^2, where k is a complex number, and he introduced a small number \epsilon to form \sqrt{k^2+\epsilon^2}=\sqrt{k+i\epsilon}\sqrt{k-i\epsilon}. The branch cut is chosen as shown in the figure attached. And the author then asserts that the branch cut so chosen is because he wants to have when \epsilon approaches zero, \sqrt{k^2}=k when Re(k)>0 and \sqrt{k^2}=-k when Re(k)<0.

    My question is how does the author get this conclusion? i.e., \sqrt{k^2}=k when Re(k)>0 and \sqrt{k^2}=-k when Re(k)<0 with this branch cut. For another example, the branch cut of \sqrt{z^2-1} would be (-1,1) because in this interval the values are different across the branch cut, while in other intervals like (-infty,-1) and (1, infty) the function is single valued.

    Thanks.
    Attached Thumbnails Attached Thumbnails I can't interpret this branch cut choice....-2014-01-03-1.45.01.png  
    Last edited by Mengqi; January 3rd 2014 at 04:08 AM.
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  2. #2
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    Re: I can't interpret this branch cut choice....

    Anybody has any clues? Or my question is not clear? Thanks.
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  3. #3
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    Re: I can't interpret this branch cut choice....

    Never mind. I have figured it out.
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