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Principal root of a complex number

  • Thread starter smize
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  • #1
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Homework Statement



I am doing a problem of a contour integral where the f(z) is z1/2. I can do most of it, but it asks specifically for the principal root. I have been having troubles finding definitively what the principal root is. Anyplace it appears online it is vague, my book doesn't speak of it, and my professor mentioned it in passing, so I have these two questions:

1. Is the principal root the branch of the root where the argument of the 2nd root is where k = 0 for it being y/2 + kπ?
2. Should I take the principal root before or after the integration? Will it make a difference?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
vela
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Could you please post the original problem? It may be easier to answer your questions in that context.
 
  • #3
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Evaluate the integral of ∫[itex]\Gamma[/itex] f(z) dz, where f(z) is the principal value of z1/2, and [itex]\Gamma[/itex] consists of the sides of the quadrilateral with vertices at the pints 1, 4i, -9, and -16i, traversed once clockwise.

I understand how to compute this for the most part. I'm just not 100% confident that I understand the principal value of z1/2, or if the principal root can be taken after integrating or if it must be taken before the integration.

Edit. Note that [itex]\Gamma[/itex] is the curve that I'm integrating on. I am bad at representing stuff using TeX.
 
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  • #4
vela
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  • #5
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Thank-you. So it is how I though.

As for the latter question, we can ignore that since I understand it now. Thank-you.
 

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