1. ## improper integral #3

Challenge Problem:

Show that $\displaystyle \int^{\infty}_{0} \frac{\cos (x^{2}) - \sin (x^{2})}{1+x^{4}} \ dx = \frac{\sqrt{2} \pi}{4e}$

2. Can we use the residue theorem?

3. Originally Posted by DeMath
Can we use the residue theorem?
You can use any method you want.

4. I wonder whether the following fact is of any use here

$\int_{0}^{\infty}e^{-y}\cos(x^2y)\;{dy} = \frac{1}{1+x^4}$

I'm not familiar with double integrals, unfortunately!

5. If you use that fact, then we have $\int^{\infty}_{0} \int^{\infty}_{0} \big(\cos(x^{2}) - \sin(x^{2}) \big) \cos (x^{2}y) e^{-y} \ dy \ dx$

But then changing the order of integration (which is justified because the original integral converges absolutely) doesn't really simply things.

The approach I had in mind was to use contour integration using a slightly unusual contour.

6. Spoiler:

Let $\displaystyle f(z) = \frac{e^{iz^{2}}}{1+z^{4}}$

And use the contour that consists of the line segment from the origin to (R,0), the arc from (R,0) to (0,iR), and the line segment back to the origin.

Interesting things happen.

7. Originally Posted by Random Variable
Challenge Problem:

Show that $\displaystyle \int^{\infty}_{0} \frac{\cos (x^{2}) - \sin (x^{2})}{1+x^{4}} \ dx = \frac{\sqrt{2} \pi}{4e}$
A method that uses the Laplace Transform: we define...

$\gamma(t)= \int_{0}^{\infty} \frac{\cos t x^{2} - \sin t x^{2}}{1+x^{4}}\ dx$ (1)

... so that is...

$\mathcal{L} \{\gamma(t)\} = \int_{0}^{\infty} e^{-s t}\ \int_{0}^{\infty} \frac{\cos t x^{2} - \sin t x^{2}}{1+x^{4}}\ dx\ dt =$

$= \int_{0}^{\infty} \frac{\mathcal{L} \{ \cos t x^{2} - \sin t x^{2} \}}{1+x^{4}}\ dx = \int_{0}^{\infty} \frac{s - x^{2}} {(1+x^{4})\ (s^{2}+x^{4})} \ dx$ (2)

To be continued in next posts...

Kind regards

$\chi$ $\sigma$

8. Performing the partial fraction expansion od the result (2) of my previous post we obtain...

$\mathcal {L} \{\gamma(t)\} = \frac{s}{s^{2}-1}\ \{\int_{0}^{\infty} \frac{dx}{1+x^{4}} - \int_{0}^{\infty} \frac{dx}{s^{2}+x^{4}} \} -$

$- \frac{1}{s^{2}-1}\ \{\int_{0}^{\infty} \frac{x^{2}}{1+x^{4}}\ dx - \int_{0}^{\infty} \frac{x^{2}}{s^{2}+x^{4}}\ dx \}$ (3)

The integrals in x can be solved in standard way with an approriate path in the complex plane obtaining...

$\int_{0}^{\infty} \frac{dx}{1+x^{4}} = \frac{\pi}{2\ \sqrt{2}}$ (4)

$\int_{0}^{\infty} \frac{dx}{s^{2}+x^{4}} = \frac{\pi}{2\ \sqrt{2}\ s^{\frac{3}{2}}}$ (5)

$\int_{0}^{\infty} \frac{x^{2}}{1+x^{4}}\ dx = \frac{\pi}{2\ \sqrt{2}}$ (6)

$\int_{0}^{\infty} \frac{x^{2}}{s^{2}+x^{4}}\ dx = \frac{\pi}{2\ \sqrt{2}\ s^{\frac{1}{2}}}$ (7)

Now is we substitute (4),(5), (6) and (7) in (3) we obtain the [incredible] result...

$\mathcal {L} \{\gamma(t)\} = \frac{\pi}{2\ \sqrt{2}}\ \frac{1}{s+1}$ (8)

... so that is...

$\gamma (t) = \frac{\pi}{2\ \sqrt{2}}\ e^{-t}$ (9)

... and finally...

$\int_{0}^{\infty} \frac{\cos x^{2} - \sin x^{2}}{1+x^{4}}\ dx = \gamma (1) = \frac{\pi}{2\ e\ \sqrt{2}}$ (10)

Kind regards

$\chi$ $\sigma$

9. Let $f(z) = \frac{e^{iz^{2}}}{1+z^{4}}$

Let $C$ be the contour consisting of the line segment from the origin to (0,R); the arc from (R,0) to (0, IR); and the line segment from (0,IR) back to the origin.

$f(z)$ has 4 simple poles at $e^{\frac{i}{4}(\pi+2 \pi n)}, \ n=0,1,2,3$

The only pole inside of the contour is $e^{\frac{i \pi}{4}}$

So we have $\int_{C} f(z) \ dz = \int^{R}_{0} f(x) \ dx + \int_{arc} f(z) \ dz + \int^{0}_{R} f(it) \ i dt = 2 \pi i \ \text{Res} [f,e^{\frac{i \pi}{4}} ]$

$\text{Res}[f,e^{\frac{i \pi}{4}}] = \lim_{z \to e^{\frac{i \pi}{4}}} \ (z - e^{\frac{i \pi}{4}}) \frac{e^{iz^{2}}}{1+z^{4}} = \frac{e^{-\frac{3 i \pi}{4}}}{4 e}$ (using L'Hospital's rule)

$= \frac{1}{4e} \Big( \frac{- 1}{\sqrt{2}} - \frac{i}{\sqrt{2}} \Big)$

$\Big| \int_{arc} f(z) \ dz \Big| = \Big| \int^{\frac{\pi}{2}}_{0} \frac{e^{i(Re^{it})^{2}}}{1+ R^{4} e^{4it}} \ i Re^{it} \ dt \Big| \le \frac{\pi}{2} \frac{e^{-R^{2} \sin 2t}}{R^{4}-1} \ R$ (ML inequality and triangle inequality)

$\le \frac{\pi}{2} e^{-R^{2}} \frac{R}{R^{4}+1}$ (since $0 \le t \le \frac{\pi}{2}$ )

therefore $\lim_{R \to \infty} \Big| \int_{arc} f(z) \ dz \Big| \le 0$

$\int^{0}_{R} f(it) \ i dt = - \int^{R}_{0} \frac{e^{i(it)^{2}}}{1+ (it)^{4}} \ i dt = - i \int^{R}_{0} \frac{e^{-it^{2}}}{1+t^{4}} \ dt$

$= -i \int^{R}_{0} \frac{\cos (t^{2}) - i \sin (t^{2}) }{1+t^{4}} \ dt = -\int^{R}_{0} \frac{\sin (t^{2}) + i \cos (t^{2})}{1+t^{4}} \ dt$

Now we have $\int_{0}^{R} \frac{ \cos (x^{2}) + i\sin (x^{2})}{1+x^{2}} \ dx + \int_{arc} f(z) \ dz - \int_{0}^{R} \frac{ \sin (x^{2}) + i\cos (x^{2})}{1+x^{4}} \ dx$

$= 2 \pi i \Big( \frac{-1}{\sqrt{2}} - \frac{i}{\sqrt{2}} \Big) = \frac{\sqrt{2} \pi}{4e} (1-i)$

Now take the limit as R goes to infinity and equate the real portions on both sides to get

$= \int^{\infty}_{0} \frac{\cos (x^{2}) - \sin (x^{2})}{1+x^{4}} \ dx = \frac{\sqrt{2} \pi}{4e}$