Contour integral (from complex analysis)

In summary, the problem involves evaluating the integral of a function f(z) along an elliptic arc g parametrized by z(t). The formula for finding the integral of f over g is given, but the attempt at finding a simplified function for f(z) was unsuccessful. The question of whether or not f(z) is analytic is brought up, as it may affect how the integral is evaluated.
  • #1
synapsis
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Homework Statement



let g denote the elliptic arc parametrized by z(t) = 2cost + 3isint, for t between 0 and pi/2 (inclusive).

Evaluate the integral of f(z) = z[sin(pi*z^2) - cos(pi*z^2)] over g.



Homework Equations



If g is determined by the function z mapping from [a,b] to C and f maps from g to C, then the integral of f over g is defined as the integral (from a to b) of f of z(t) times z'(t).

(sorry for writing the equations out in words, I don't have any formatting software)



The Attempt at a Solution



I started by finding z'(t) = -2sint + 3icost and attempting to find f(z(t)), but I got a really complicated function and at that point I figured I must be going about it the wrong way.

I tried to find an identity that would allow me to simplifiy f(z) but I couldn't find anything.

At this point I really have no idea how to proceed.
 
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  • #2
If f(z) is analytic, what do you know about its integral along a contour between two points?
 
  • #3
hmm...I don't know. It doesn't say so in the problem. Is that something I should be able to recognize?
 
  • #4
Yes, or at least, it's something they want you to learn. I imagine that's the point of this problem.
 

1. What is a contour integral?

A contour integral is a type of line integral that is calculated along a specific path or contour in a complex plane. It involves integrating a complex-valued function over a specific curve or closed loop in the complex plane.

2. What is the significance of contour integrals in complex analysis?

Contour integrals are important in complex analysis because they allow us to evaluate complex-valued functions along specific paths in the complex plane. They also help us to understand the behavior of complex functions over specific regions in the complex plane.

3. How is a contour integral calculated?

To calculate a contour integral, we first need to parameterize the contour or path in the complex plane. Then, we substitute the parameterized form of the contour into the complex-valued function and integrate it with respect to the parameter. Finally, we evaluate the resulting integral to find the value of the contour integral.

4. What is the difference between a contour integral and a line integral?

The main difference between a contour integral and a line integral is that a contour integral is calculated along a specific path in the complex plane, whereas a line integral is calculated along a specific curve in the real plane. Additionally, contour integrals involve complex-valued functions, while line integrals can involve real-valued functions.

5. In what fields of science and mathematics are contour integrals commonly used?

Contour integrals have many applications in science and mathematics, including in physics, engineering, and complex analysis. They are commonly used in the study of fluid dynamics, electromagnetism, and quantum mechanics, as well as in the evaluation of complex functions and path integrals in mathematics.

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