Principal root of a complex number

In summary, the conversation is about a problem involving a contour integral with a function of z1/2. The main question is about the principal root and whether it should be taken before or after the integration. The problem also involves a quadrilateral with specific points as vertices and the curve being integrated on. The conversation also discusses the branch cut and the understanding of the principal value of z1/2.
  • #1
smize
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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?
 
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  • #2
Could you please post the original problem? It may be easier to answer your questions in that context.
 
  • #3
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.
 
Last edited:
  • #5
Thank-you. So it is how I though.

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

1. What is the principal root of a complex number?

The principal root of a complex number is the square root of the complex number that has the largest argument (angle) between 0 and 2π. It is also known as the principal square root.

2. How do you find the principal root of a complex number?

To find the principal root of a complex number, we can use the polar form of the complex number. We first convert the complex number into polar form, then take the square root of the magnitude and divide the argument by 2. The resulting values are the real and imaginary parts of the principal root.

3. Can the principal root of a complex number be a complex number itself?

Yes, the principal root of a complex number can be a complex number itself. This is because the square root of a negative number results in a complex number, and the principal root is the square root with the largest argument.

4. Is the principal root of a complex number unique?

Yes, the principal root of a complex number is unique. This is because the principal root is defined as the square root with the largest argument, and the argument of a complex number is always unique between 0 and 2π.

5. What is the significance of the principal root of a complex number?

The principal root of a complex number is significant because it helps us determine the other roots of a complex number. By using the principal root, we can find the other roots by adding multiples of 2π to the argument of the principal root.

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