Contour Integration: Rules and Limits

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In summary, when doing a trigonometric type contour integration with limits of integration between -Pi and Pi, the integrand cannot be any function. The number and type of singularities it has around the region of integration can cause the integral to be ill-defined. Additionally, the original limits should not affect the result of the integral, as long as the correct counter is defined. However, if f(z) does not have any singularities, it can be any function.
  • #1
jenga42
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I'm trying to do a trigonometric type contour integration... one with limits of integration between -Pi and Pi. Is there a rule about what the integrand has to look like?

Eg can I do this (I've converted it into z's):

int( f(z)/( (z-z1)^2 (z-z2)^2 ) ,z)

...can f(z) be any function?

And is it okay for the original limits to be from -Pi to Pi rather than the normal 0 to 2*Pi

Thanks!
 
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  • #2
f(z) can't be any function, the number and type of singularities it has around the region of integration can cause the integral to be ill-defined. Also, the limits should not matter, but if this is a contour integral, what is the counter? You define bounds like a normal integral.
 
  • #3
Thanks for replying...

I should have written "if f(z) doesn't have any singularities, can it be any function?" .. can it?

Thanks
 

1. What is contour integration?

Contour integration is a mathematical technique used to evaluate integrals along a given path, known as a contour, in the complex plane. It is an extension of the fundamental theorem of calculus to complex-valued functions.

2. What are the rules for contour integration?

The basic rules for contour integration include the use of Cauchy's integral formula, the Cauchy-Goursat theorem, and the Cauchy integral theorem. These rules allow for the evaluation of integrals using the properties of analytic functions and their derivatives.

3. Why is contour integration useful?

Contour integration is useful for solving difficult integrals that cannot be evaluated using traditional methods. It allows for the use of complex analysis techniques to simplify the integration process and obtain more accurate results.

4. What are the limits of contour integration?

The main limitation of contour integration is that it can only be applied to functions that are analytic within the contour. This means that the function must be differentiable at every point within the contour. Additionally, the contour must be a closed path and cannot contain any singularities.

5. How is contour integration applied in real-world problems?

Contour integration has various applications in physics, engineering, and mathematics. It can be used to solve problems in fluid dynamics, electromagnetism, and quantum mechanics, among others. It also has practical applications in signal processing and image reconstruction.

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