# Principal root of a complex number

1. Sep 29, 2013

### smize

1. The problem statement, all variables and given/known data

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?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 29, 2013

### vela

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

3. Sep 29, 2013

### smize

Evaluate the integral of ∫$\Gamma$ f(z) dz, where f(z) is the principal value of z1/2, 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 z1/2, 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.

Last edited: Sep 29, 2013
4. Sep 30, 2013

### vela

Staff Emeritus
5. Sep 30, 2013

### 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.