Complex analysis question

• ehrenfest
In summary, the problem of evaluating the integral \int_{\gamma} \sqrt {z} dz on the upper half of the unit circle is considered to be ambiguous because it does not specify which branch of the complex square root function to use. However, it is implied that one should choose a branch for which the integral makes sense. It is also noted that the integral must be defined in order to be evaluated. Additionally, it is mentioned that the independence of the integral on the chosen branch depends on the function having an analytic antiderivative in a region containing the path. The specific proof for this is not remembered by the speaker.

Homework Statement

Evaluate $$\int_{\gamma} \sqrt {z} dz$$ where $$\gamma$$ is the upper half of the unit circle.

I contend that this problem does not make sense i.e it is ambiguous because they did not tell us specifically what branch of the complex square root function to use. Am I right?

The Attempt at a Solution

Well, the integral isn't defined if the function fails to be continuous on the path. So, while you could say, "It's too ambiguous," it's pretty clear that they mean, "Pick a branch for which this makes sense." Now, if the integral winds up depending on the branch you choose for which it makes sense, maybe you've got a problem. But I'm pretty sure it wouldn't.

Though don't quote me on that. I suck at this stuff for some reason.

Anyone know how to prove that the integral is independent of the branch if the branch makes sense?

Well, that has to do with the function having analytic antiderivative in a region containing the path. Can't remember the proof off the top of my head though.

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers, which are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. It involves the study of functions of complex variables and their properties.

2. What is the difference between real analysis and complex analysis?

The main difference between real analysis and complex analysis is that real analysis deals with real numbers and their functions, while complex analysis deals with complex numbers and their functions. Real analysis is focused on real-valued functions, while complex analysis is focused on complex-valued functions.

3. What are some applications of complex analysis?

Complex analysis has numerous applications in different fields, such as physics, engineering, and finance. It is used to solve problems involving electric circuits, fluid dynamics, signal processing, and more. In finance, complex analysis is used to study the behavior of stock market prices and to develop mathematical models for predicting market trends.

4. What are some important theorems in complex analysis?

Some important theorems in complex analysis include the Cauchy-Riemann equations, Cauchy's integral theorem, Cauchy's integral formula, and the residue theorem. These theorems are fundamental in understanding the properties of complex functions and their behavior.

5. How is complex analysis used in engineering?

Complex analysis is used in engineering to solve problems involving electric circuits, signal processing, and control systems. It is also used in the design and analysis of structures and materials, as well as in the study of fluid dynamics and heat transfer. In addition, complex analysis is used in the development of numerical methods for solving engineering problems.