Complex Analysis: Integrating rational functions

In summary, the conversation discusses integrating rational functions over the unit circle and the use of the substitution z = e^(it) to simplify the integration process. The authors argue that this substitution is allowed because it makes the integration process easier, but there is no clear explanation as to why it works.
  • #1
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Homework Statement


Hi all.

My question has to do with integrating rational functions over the unit circle. My example is taken from here (page 2-3): http://www.maths.mq.edu.au/%7Ewchen/lnicafolder/ica11.pdf

We wish to integrate the following

[tex]
\int_0^{2\pi } {\frac{{d\theta }}{{a + \cos \theta }}}.
[/tex]

According to the .pdf, we integrate along a unit circle, and we define [itex] z = e^{i\theta}[/itex]. When rewriting the integrating using z, we find that the integrand has to poles: One inside the unit circle and one outside. When we use the residue theorem, we only use the pole inside the unit circle.

My question: How are we even allowed just to say: "We choose only to integrate along a unit circle, and thus we only look at poles inside this circle"? If I claim that we should integrate along a circle big enough to include all poles, then who can say argument against my claim?

I hope you understand me. Niles.
 
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  • #2
The reasoning is not "we must integrate along a unit circle, therefore we substitute z=e^(it)", but "let's substitute z=e^(it), then - since t ranges from 0 to 2*pi - we are integrating along a unit circle".
If you want to substitue z=Re^(it), so that you integrate along a non-unit circle, you could try that, but the question is whether that will work. The substitution z=e^(it) is easy to work with.
 
  • #3
Landau said:
If you want to substitue z=Re^(it), so that you integrate along a non-unit circle, you could try that, but the question is whether that will work. The substitution z=e^(it) is easy to work with.

I have not yet seen a proof of that substituting z = exp(it) works as well, only that it apparently makes things easy.
 
  • #4
It turns out to work as shown in the pdf you're referring to, right?
 
  • #5
In the PDF the author uses z = exp(it), and I have seen this done in other notes on this topic as well. But none of the authors explain why they are allowed to do this other than it works out in the end.
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the properties and behavior of complex numbers, which are numbers that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. It involves studying functions and their derivatives and integrals in the complex plane.

2. What is a rational function?

A rational function is a function that can be written as the ratio of two polynomial functions. In other words, it is a fraction where the numerator and denominator are both polynomials. Examples of rational functions are f(x) = (x+1)/(x^2+3x+2) and g(x) = (x^3 + 2x^2 - 5)/(x^2 + 1).

3. How do you integrate a rational function?

To integrate a rational function, we use the method of partial fractions. This involves breaking the rational function into simpler fractions, each with a distinct denominator, and then integrating each term separately. The final result is the sum of the integrals of the individual terms.

4. What is the importance of integrating rational functions in complex analysis?

Integrating rational functions is important in complex analysis because it allows us to calculate the area under a curve in the complex plane. This is essential in many applications, such as calculating work done by a force in physics or finding the volume of a complex object in engineering.

5. Are there any special techniques for integrating rational functions in complex analysis?

Yes, there are several techniques that can be used to make integrating rational functions easier. These include using trigonometric substitutions, completing the square, and applying the residue theorem, which involves using complex contour integrals to evaluate the integral of a rational function. It is important to choose the appropriate technique based on the form of the rational function being integrated.

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