Improper complex integrals--residue

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In summary, the first integral can be solved by using the even function property and the second integral can be solved using complex analysis methods, resulting in a sum of the residues which equals pi*sqrt(2)/2.
  • #1
Dustinsfl
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Given this:
$$
\int_0^{\infty}\frac{x^2}{x^6 + 1}dx = \frac{\pi}{6}
$$
Can I do this:
$$
\int_0^{\infty}\frac{x^2}{x^6 + 1}dx = \frac{1}{2}\int_{-\infty}^{\infty}\frac{x^2}{x^6 + 1}dx
$$
and solve the integral like this
$$
\int_{-\infty}^{\infty}\frac{x^2}{x^4 + 1}dx = 2i\pi\sum_{z \ \text{upper half}}\text{Res}_{z}f = \frac{\pi\sqrt{2}}{2}
$$
 
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  • #2
dwsmith said:
Given this:
$$
\int_0^{\infty}\frac{x^2}{x^6 + 1}dx = \frac{\pi}{6}
$$
Can I do this:
$$
\int_0^{\infty}\frac{x^2}{x^6 + 1}dx = \frac{1}{2}\int_{-\infty}^{\infty}\frac{x^2}{x^6 + 1}dx
$$
Yes because the integral is convergent and you're using the way back of the even function when having a symmetric interval.
I never learned well about computing integrals by using complex analysis methods so I'll let another one which may confirm your procedure.
 
  • #3
Krizalid said:
Yes because the integral is convergent and you're using the way back of the even function when having a symmetric interval.
I never learned well about computing integrals by using complex analysis methods so I'll let another one which may confirm your procedure.

It works because it is an even function. I actually just shown the integral is pi/6
 

FAQ: Improper complex integrals--residue

1. What is an improper complex integral?

An improper complex integral is an integral that involves complex numbers and may not have a definite value due to singularities or infinite limits of integration.

2. What is a residue in the context of complex integration?

In complex analysis, a residue is the coefficient of the term with a negative power in the Laurent series expansion of a complex function around a singularity. It is used to evaluate improper complex integrals.

3. How do you calculate residues?

To calculate residues, you need to find the Laurent series expansion of the complex function around the singularity. Then, the residue is equal to the coefficient of the term with a negative power.

4. When do we use the residue theorem to evaluate improper complex integrals?

The residue theorem can be used to evaluate improper complex integrals when the integrand has simple poles, which are isolated singularities with a first-order pole. It cannot be used for essential singularities or poles of higher order.

5. Can the residue theorem be used for integrals with multiple poles?

Yes, the residue theorem can be used for integrals with multiple poles. This is because the residues of each pole can be evaluated separately and then added together to get the final result.

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