Math. physics

1. Dec 5, 2011

matematikuvol

1. The problem statement, all variables and given/known data

Prove

$$\sqrt{\frac{2}{\pi}}\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=t^{-\frac{1}{2}}$$

and use that to solve

$$\int^{\infty}_0\cos y^2dy$$

Is this good way to try to prove?

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data
Multiplicate both sides with $$\cos x'tdt$$ and integrate from zero to $$\infty$$

$$\sqrt{\frac{2}{\pi}}\int^{\infty}_0dt\cos (x't)\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=\int^{\infty}_0dt\cos (x't)t^{-\frac{1}{2}}=\sqrt{\frac{2}{\pi}}\int^{\infty}_0dxx^{-\frac{1}{2}}\int^{\infty}_0dt\cos (x't)\cos (xt)dx$$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Dec 5, 2011

matematikuvol

Suppose that we know

$$\sqrt{\frac{2}{\pi}}\int^{\infty}_0\cos(xt)x^{-\frac{1}{2}}dx=t^{-\frac{1}{2}}$$

without proving. How to calculate then

$$\int^{\infty}_0\cos x^2dx$$

Last edited: Dec 5, 2011
3. Dec 5, 2011

crimsonidol

you need to use residue calculus.
if you can go to library look at hildebrand advanced calculus for applications, under the intended contours you will see how to use cauchy's principle and then you'll get gamma functions.

4. Dec 5, 2011

matematikuvol

For which part of problem. This is problem from Arfken, Weber.

5. Dec 5, 2011

crimsonidol

you can do the proof and also find part b when you understand the first part I assume.look hildebrand page 561 to be exact

6. Dec 5, 2011

matematikuvol

$$\int^{\infty}_0\frac{\cos x}{x^{1-m}}dx=\Gamma(m)\cos (\frac{m\pi}{2})$$

$$\int^{\infty}_0\frac{\cos x}{x^{1-\frac{1}{2}}}dx=\Gamma(\frac{1}{2})\cos (\frac{\frac{1}{2}\pi}{2})=\sqrt{\pi}\frac{\sqrt{2}}{2}=\sqrt{\frac{\pi}{2}}$$

I don't see solution :(

7. Dec 5, 2011

crimsonidol

you need to look at contour integration and use xt instead of x there. By using residue and appropriate contour you'll be able to find t^-1/2

8. Dec 5, 2011

matematikuvol

Ok. Thanks. And what then. When I prove first part, how can I calculate integral $$\int^{\infty}_0\cos x^2dx$$?

9. Dec 5, 2011

crimsonidol

10. Dec 5, 2011

dextercioby

Take t=1 in what you have proven already then do a simple substitution to get rid of the sqrt.

11. Dec 6, 2011

matematikuvol

Thanks a lot! :)