# Complex contour integral of square-root

• Yeggoua
In summary: Just keep track of the signs that you get when you plug in the endpoints.In summary, the student is asking for help with evaluating a complex contour integral involving a square root function. They are unsure how to handle the different branches and are looking for assistance with the problem.
Yeggoua
For my homework I am told: "Evaluate $z^(1/2)dz around the indicated not necessarily circular closed contour C = C1+C2. (C1 is above the x axis, C2 below, both passing counter-clockwise and through the points (3,0) and (-3,0)). Use the branch r>0, -pi/2 < theta < 3*pi/2 for C1, and the branch r>0, pi/2 < theta < 5*pi/2 for C2." I am unsure how to approach this and would appreciate help. I get that I need to take the anti-derivative to get 2/3*z^(3/2) and possibly convert to polar form, but am unsure how to handle the different 'branches' to find an answer. Thanks Yeggoua said: For my homework I am told: "Evaluate$z^(1/2)dz around the indicated not necessarily circular closed contour C = C1+C2. (C1 is above the x axis, C2 below, both passing counter-clockwise and through the points (3,0) and (-3,0)). Use the branch r>0, -pi/2 < theta < 3*pi/2 for C1, and the branch r>0, pi/2 < theta < 5*pi/2 for C2."

I am unsure how to approach this and would appreciate help. I get that I need to take the anti-derivative to get 2/3*z^(3/2) and possibly convert to polar form, but am unsure how to handle the different 'branches' to find an answer.

The bit about the branches is that you need to choose which square root you use when you're integrating. (Recall that every number has two square roots.) That's really all that the 'branches' stuff is about.

You should be able to integrate along the two pieces seperately without any special problems.

## 1. What is a complex contour integral?

A complex contour integral is a mathematical concept that involves integrating a function along a specific path or curve in the complex plane. Unlike real integrals, which are calculated on a straight line, complex contour integrals take into account the complex structure of the function being integrated.

## 2. How do you calculate a complex contour integral?

To calculate a complex contour integral, you need to first determine the function being integrated and the path or curve along which the integration will be performed. Then, you can use techniques such as the Cauchy integral theorem or the Cauchy integral formula to evaluate the integral. These techniques involve breaking the integral into smaller, simpler parts and using properties of complex numbers to solve them.

## 3. What is the significance of the square-root in a complex contour integral?

The square-root in a complex contour integral is significant because it represents the branch cut of the function being integrated. This means that the complex plane is divided into two parts, and the function is evaluated differently on each part. The square-root helps to specify which part of the function is being evaluated at each point on the curve or path.

## 4. What are some applications of complex contour integrals?

Complex contour integrals have many applications in mathematics, physics, and engineering. They are commonly used in the study of complex analysis, which has applications in fields such as fluid dynamics, electromagnetism, and signal processing. They are also used in the evaluation of complex functions and in solving differential equations.

## 5. Are there any challenges in calculating a complex contour integral of a square-root?

Yes, there can be challenges in calculating a complex contour integral of a square-root. One challenge is choosing the appropriate path or curve for the integration, as different paths can lead to different results. Another challenge is dealing with singularities, which are points where the function being integrated becomes undefined or infinite. These challenges can require advanced techniques and careful consideration to accurately evaluate the integral.

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