Evaluating Complex Integrals: From Pi to i & Around Unit Circle & Square

In summary, the conversation revolves around evaluating the integral of e^(iz)dz with bounds from pi to i using different methods. The first method involves evaluating it along a straight line joining the limits and along segments of the coordinate axes joining the limits. The second method involves estimating the integral using a given inequality and comparing it with the actual value. The conversation also touches on converting z to x+iy and z conjugate to x-iy. Finally, there is a discussion on evaluating the same integral taken in the positive sense about a square with given coordinates. A possible parametrization for the path from pi to i is suggested.
  • #1
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Homework Statement



integral(e^(iz)dz) with bounds from pi to i.
I need to evaluate using : along the straight line joining the limits, along segments of the coordinate axes joining the limits, and estimate using integral <= L*M and compare the estimates with the values.

I am totally confused on how to do this. Do I convert z to x+iy and do the bounds change?



I have another integral with z conjugate about the unit circle. I think I have an idea on this one on converting to x-iy and seeing that this will equal 2ipi.



I am also asked to evaluate this same integral taken in the positive sense about the square with x=+-1, y=+-1. I know this involves 4 segments, but my problem is finding this 4 segments. I know the answer is 8i, but how to get there?
 
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  • #2
You can parametrize the path from pi to i, e.g by taking:

z(t) = pi + (i-pi)t

The starting point is then at t = 0 and the end point at t = 1.

integral f(z)dz along the path then means:

f(z(t))d(z(t)) from t = 0 to t = 1.

We have dz = (i-pi) dt

and

f(z(t)) = exp[i pi - (i pi + 1)t] = -exp[-(i pi + 1)t]
 
  • #3
That makes a lot of sense actually.
 

1. What is a complex integral and how is it different from a regular integral?

A complex integral is an integral that involves complex numbers, whereas a regular integral only involves real numbers. In a complex integral, the function being integrated can have both real and imaginary parts.

2. How do you evaluate a complex integral?

The process of evaluating a complex integral involves using techniques such as Cauchy's integral theorem, residue theorem, and contour integration. These techniques involve manipulating the complex integral to make it easier to solve, such as by changing the path of integration or using Cauchy's integral formula.

3. What is the significance of the unit circle and square in evaluating complex integrals?

The unit circle and square are commonly used as contours in complex integration. The unit circle has a radius of 1 and is used to evaluate integrals around the origin, while the unit square is used to evaluate integrals along the real and imaginary axes. These contours make it easier to evaluate complex integrals using techniques such as Cauchy's integral theorem.

4. How does the value of pi and the imaginary unit i come into play when evaluating complex integrals?

The value of pi is often used in evaluating complex integrals as it is a key factor in Cauchy's integral formula. The imaginary unit i is also used in complex integrals to represent the square root of -1, which is necessary for manipulating and solving integrals with complex numbers.

5. Can complex integrals be used in real-world applications?

Yes, complex integrals have many real-world applications, particularly in fields such as engineering, physics, and economics. They can be used to solve problems involving oscillations, electrical circuits, fluid dynamics, and more. Complex integrals are also commonly used in signal processing and control systems.

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