Principal root of a complex number

[smize]

Homework Statement

I am doing a problem of a contour integral where the f(z) is z^1/2. 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?

 

[vela]

Could you please post the original problem? It may be easier to answer your questions in that context.

 

[smize]

Evaluate the integral of ∫_\Gamma f(z) dz, where f(z) is the principal value of z^1/2, and \Gamma 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 z^1/2, or if the principal root can be taken after integrating or if it must be taken before the integration.

Edit. Note that \Gamma is the curve that I'm integrating on. I am bad at representing stuff using TeX.

 

[vela]

See http://en.wikipedia.org/wiki/Square_root#Principal_square_root_of_a_complex_number

I'd say the problem is telling you to put the branch cut along the -x axis.

I'm not sure what you mean by "take the principal root before or after the integration."

 

[smize]

Thank-you. So it is how I though.

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