Determining Suitable Contour for $\int_0^{\infty} \dfrac{\log (x) }{x^2 +1 } dx$

  • MHB
  • Thread starter Amer
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In summary: So the contour $\left(\oint_{C_R}+\oint_{c_r} \right)f(z)\,dz$ is valid.In summary, the suitable contour for the integral $\displaystyle \int_0^{\infty} \dfrac{\log (x) }{x^2 +1 } dx$ is a combination of a semicircle in the clockwise direction, rays emanating from the endpoints of the semicircle, and a circle centered at the origin with a large radius. This contour is used because it allows for the integration of the function $\frac{\log z}{z^2+1}$, where $z$ is a complex variable, and results in the integral being equal to $2\
  • #1
Amer
259
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what is the suitable contour for the integral
$\displaystyle \int_0^{\infty} \dfrac{\log (x) }{x^2 +1 } dx $

and why ? or what is the method to determine a good one?
 
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  • #2
Amer said:
what is the suitable contour for the integral
$\displaystyle \int_0^{\infty} \dfrac{\log (x) }{x^2 +1 } dx $

and why ? or what is the method to determine a good one?

Let $\varepsilon > 0$ be small and $R>0$ be large.
1) Start at $i\varepsilon$, in the clockwise direction draw semicircle $C$ centered at $0$ that ends at $-i\varepsilon$.
2) Draw rays emanating at $\pm i\varepsilon$, endpoints of $C$, to the left, so $-t\pm i\varepsilon $ for $t\geq 0$.
3) Draw circle $\Gamma$ centered at $0$ of radius $R$.
4) The rays intersect $\Gamma$ at points $\alpha,\beta$ with $\text{Im}(\alpha) > \text{Im}(\beta)$.

Now define $\Gamma_1$ as contour starting at $\beta$, travel counterclockwise along $\Gamma$ until reaching $\alpha$. Travel linearly along ray until reaching $i\varepsilon$. Then move clockwise along $C$ until reaching $-i\varepsilon$. Then travel linearly along ray until returning back to $\alpha$.

Define the function,
$$ f(z) = \frac{\log z}{z^2 + 1} $$
Where $\log z$ is the principal logarithm and integrate,
$$ \oint_{\Gamma_1} f(z) ~ dz $$
 
  • #3
Amer said:
what is the suitable contour for the integral
$\displaystyle \int_0^{\infty} \dfrac{\log (x) }{x^2 +1 } dx $

and why ? or what is the method to determine a good one?

In...

http://mathhelpboards.com/calculus-10/improper-integral-involving-ln-6103-post27786.html#post27786

... in elementary fashion it has been demonstrated that...

$\displaystyle \int_{0}^{\infty} \frac{\ln x}{1 + x^{2}}\ dx = 0\ (1)$

... but it is also been demonstrated that no suitable contour of integration in the complex plane for the integral (1) exists... Kind regards $\chi$ $\sigma$
 
  • #4
The best way of computing that integral is to rather use a substitution $x\mapsto 1/x$.

Let $I = \int_0^{\infty} \frac{\log x}{x^2+1} ~ dx$. Now, let $y = \frac{1}{x}$, so that $y' = -\frac{1}{x^2}$ and $x = \frac{1}{y}$. Rewrite,
$$ \frac{\log x}{x^2+1} = \frac{\log x}{x^2+1} \cdot \left( -\frac{1}{x^2} \right) \cdot \left( -x^2 \right) = \frac{ \log \frac{1}{y}}{ \frac{1}{y^2} + 1 } \cdot y' \cdot \left( -\frac{1}{y^2} \right) = \frac{\log y}{y^2+1} \cdot y' $$
Futhermore, $y(0) = \infty$ and $y(\infty) = 0$, so that,
$$ I = \int_0^{\infty} \frac{\log x}{x^2+1} ~ dx = \int_{\infty}^{0} \frac{\log y}{y^2+1} ~ dy = - I $$
This implies that $I=0$.
 
  • #5
ThePerfectHacker said:
Let $\varepsilon > 0$ be small and $R>0$ be large.
1) Start at $i\varepsilon$, in the clockwise direction draw semicircle $C$ centered at $0$ that ends at $-i\varepsilon$.
2) Draw rays emanating at $\pm i\varepsilon$, endpoints of $C$, to the left, so $-t\pm i\varepsilon $ for $t\geq 0$.
3) Draw circle $\Gamma$ centered at $0$ of radius $R$.
4) The rays intersect $\Gamma$ at points $\alpha,\beta$ with $\text{Im}(\alpha) > \text{Im}(\beta)$.

Now define $\Gamma_1$ as contour starting at $\beta$, travel counterclockwise along $\Gamma$ until reaching $\alpha$. Travel linearly along ray until reaching $i\varepsilon$. Then move clockwise along $C$ until reaching $-i\varepsilon$. Then travel linearly along ray until returning back to $\alpha$.

Define the function,
$$ f(z) = \frac{\log z}{z^2 + 1} $$
Where $\log z$ is the principal logarithm and integrate,
$$ \oint_{\Gamma_1} f(z) ~ dz $$

That combination of function and contour won't work. The log term will vanish.

You either need to consider $ \displaystyle f(z) = \frac{\log^{2} z}{1+z^{2}} $, or you could move the branch cut from the negative real axis to the negative imaginary axis and integrate around a contour that consists of the real axis (indented at the origin) and the upper half of the circle $|z|=R$.
 
  • #6
If we integrate the function

$$f=\frac{\log^2(z)}{z^2+1}$$

along the contour

View attachment 2411

$$\left(\oint_{C_R}+\oint_{c_r} \right)f(z)\,dz + \int^R_r \frac{\log^2(x)}{x^2+1}\,dx-\int^{R}_r\frac{(\log(x)+2\pi i)^2}{x^2+1}\,dx =2\pi i \sum_{i=1}^2\text{Res}(f,z_i)$$

Taking $R\to \infty$ and $r \to 0$

Then we have

$$-4\pi i\int^\infty_0 \frac{\log(x)}{x^2+1}\,dx+4\pi^2\int^{\infty}_0\frac{1}{x^2+1}\,dx =2\pi i \sum_{i=1}^2\text{Res}\left(\frac{\log^2(z)}{z^2+1},\pm i \right)$$

$$\sum_{i=1}^2\text{Res}\left(\frac{\log^2(z)}{z^2+1},\pm i \right)=\left(\frac{\log^2(i)}{2i} -\frac{\log(-i)^2}{2i}\right)=-\frac{\pi^2}{8i}+\frac{9\pi^2}{8i}=-i\pi^2$$

$$-4\pi i\int^\infty_0 \frac{\log(x)}{x^2+1}\,dx+4\pi^2\int^{\infty}_0\frac{1}{x^2+1}\,dx =2\pi^3$$

Hence we have that

$$\int^{\infty}_0\frac{1}{x^2+1}\,dx=\frac{\pi}{2}$$

$$\int^\infty_0 \frac{\log(x)}{x^2+1}\,dx=0$$
 

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FAQ: Determining Suitable Contour for $\int_0^{\infty} \dfrac{\log (x) }{x^2 +1 } dx$

What is the purpose of determining a suitable contour for the integral?

The purpose of determining a suitable contour for the integral is to find a path in the complex plane that allows us to evaluate the integral without encountering any singularities or discontinuities. This contour should also ensure that the integral is convergent.

How do you determine a suitable contour for the integral?

To determine a suitable contour for the integral, we first need to identify any singularities or discontinuities in the integrand. Then, we can choose a contour that avoids these points and also ensures convergence of the integral. This may involve using techniques such as residues or Jordan's lemma.

Why is it important to choose the right contour for the integral?

Choosing the right contour for the integral is important because it allows us to accurately evaluate the integral and obtain the correct result. If the contour is not suitable, it may lead to incorrect or divergent results.

Can the same contour be used for all integrals?

No, the same contour cannot be used for all integrals. The choice of contour depends on the specific function being integrated and the location of its singularities. Different integrals may require different contours to ensure convergence and accurate evaluation.

Are there any general rules for choosing a contour for an integral?

There are no general rules for choosing a contour for an integral. The best approach is to analyze the function and its singularities, and then use techniques such as residues or Jordan's lemma to determine a suitable contour. Experience and knowledge of complex analysis can also help in choosing an appropriate contour.

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