Quick complex analysis (integration) question

• indigogirl
In summary, The integral from -1 to 1 of z^i is equal to (1-i)(1+exp(-pi)/2) where the path of integration is any contour above the real axis from z=-1 to z=1. This can be shown by using the fact that z^i=exp(i log(z)) and choosing a sample contour, or by using the antiderivative of z^i=z^(i+1)/(i+1) since there are no branch cuts involved.
indigogirl
I want to show that the integral from -1 to 1 of z^i = (1-i)(1+exp(-pi)/2

where the path of integration is any contour from z=-1 to z=1 that lies above the real axis (except for its endpoints).

So, I know that z^i=exp(i log(z)) and the problem states that |z|>0, and arg(z) is between -pi/2 and 3pi/2. But we didn't study how to integrate z^(complex number) in class, and I"m really confused on how to do this.

So, how do I integrate this?

You could pick a sample contour like z=exp(it) for t=0 to pi. Then log(z)=it, dz=i*exp(it)*dt. Change it into a t integration. You could also just use the antiderivative of z^i=z^(i+1)/(i+1) since you aren't passing over any branch cuts.

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It involves the analysis of complex-valued functions, which are functions with inputs and outputs in the complex number system.

2. What is integration in complex analysis?

Integration in complex analysis is the process of finding the area under a curve in the complex plane. It is similar to integration in real analysis, but involves complex-valued functions and complex numbers as inputs and outputs.

3. How is integration different in complex analysis compared to real analysis?

The main difference between integration in complex analysis and real analysis is that in complex analysis, the path of integration is taken into account. This means that the integral can have different values depending on the path taken, unlike in real analysis where the value of the integral is independent of the path.

4. What is the Cauchy integral formula?

The Cauchy integral formula is a fundamental theorem in complex analysis that relates the value of a complex-valued function at a point inside a closed curve to the values of the function along the curve. It is used to evaluate integrals in complex analysis.

5. What is the importance of complex analysis in science and engineering?

Complex analysis has many applications in science and engineering, particularly in the fields of physics, chemistry, and electrical engineering. It is used to study physical phenomena and solve problems involving complex variables, such as in fluid dynamics, electrical circuits, and quantum mechanics.

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