# Are residues useful for proper integrals?

• jinawee
In summary, residues are useful when we are trying to solve improper integrals, because the Cauchy principal value will be the sum of residues inside the path taken.
jinawee
Calculating residues are useful when we are trying to solve some improper integral, because the Cauchy principal value will be the sum of residues inside the path taken (if the integral along the complex path tends towards 0).

When we have a proper integral of trigonometric functions, this is useful too.

But in general, are residues useful calculating proper integral?

It would be nice if I could apply the power of residues to evaluate an integral involving rational functions.

Although residues were developed primarily for evaluating complex integrals, they can be used to evaluate the integrals of real functions as well. Any decent text on complex integration should include a section on this topic.

This paper has some examples:

http://people.math.gatech.edu/~cain/winter99/supplement.pdf

SteamKing said:
Although residues were developed primarily for evaluating complex integrals, they can be used to evaluate the integrals of real functions as well. Any decent text on complex integration should include a section on this topic.

This paper has some examples:

http://people.math.gatech.edu/~cain/winter99/supplement.pdf

I'm aware of that use, but I'm asking for definite integrals.

In general,

$$\int^{R}_{-R}f(x)dx+\int_{C_R} f(z)dz=2\pi i\sum \mathrm{Res}$$

For example,

$$\int^{10}_{-10}\frac{x^2}{x^6+1}dx=\frac{\pi}{3}-\int_{C_R} \frac{z^2}{z^6+1}dz$$

Where CR is a circular contour from 10 to -10. The problem is that the contour integral is not zero, so we would have something like this:

$$\int^{10}_{-10}\frac{x^2}{x^6+1}dx=\frac{\pi}{3}-\int_{C_R} \frac{x^2}{x^6+1}dz=\frac{\pi}{3}-i \int^{\pi}_{0} \frac{10^3 e^{i3\theta}}{10^6 e^{i6\theta}+1} d \theta$$

This method seems to complicate things. But is there any case where it's useful?

It's not clear why you want to use residues for this particular problem. It can be handled relatively simply by using substitution.

SteamKing said:
It's not clear why you want to use residues for this particular problem. It can be handled relatively simply by using substitution.

Sorry, I didn't explain myself correctly. It was just an example.

I just want to know if it could be useful for other definite integral that doesn't involve trigonometric functions (eliptic, hyperbolic...).

wikipedia http://en.wikipedia.org/wiki/Methods_of_contour_integration gives

$$\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx = \frac{\pi}{2\sqrt{2}} \left(17 - 40^{\frac{3}{4}} \right)$$

Periodic and improper examples are more common because closing the contour trades one integral for another. In order for the trade to be an improvement we need an integral to be easily found, zero, or expressible in terms of another. This happens often for periodic and improper examples, less so in other cases. Notice the above example takes advantage of another opportunity, branch cuts.

1 person

## 1. What are residues in the context of proper integrals?

Residues are the values of a complex function at its singular points, which are the points where the function is not defined or is undefined. In the context of proper integrals, residues are used to evaluate integrals that cannot be solved using traditional methods.

## 2. How are residues used to solve proper integrals?

Residues are used in the method of residue integration, which involves finding the singular points of a complex function and calculating the residues at those points. The residues are then used in a formula to evaluate the integral.

## 3. Can residues be used for all types of proper integrals?

No, residues can only be used for proper integrals that involve complex functions with singular points. They cannot be used for integrals that do not have singular points, such as simple polynomials.

## 4. What are the advantages of using residues for proper integrals?

Using residues allows for the evaluation of integrals that cannot be solved using traditional methods, making it a powerful and versatile tool for solving complex integrals. It also provides a more efficient method for evaluating certain types of integrals.

## 5. Are residues always useful for solving proper integrals?

No, there are cases where the use of residues may not be the most efficient or effective method for solving a proper integral. It is important to consider other methods and techniques before deciding to use residues.

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