MHB Fresnel Integrals: Unsolved Question from MHF

[Fernando Revilla]

I quote a unsolved question posted in MHF by user poorbutttryagin on February 5th, 2013.

I read 'functions of one complex variable by Conway'

186pg, 7.7. Prove that int_0^inf sin(t^2) dt = sqrt(pi/8)

What is the starting point?

Any comment or hint is welcomed !

Thanks !

 

[Fernando Revilla]

Have a look at the pdf here:

http://www.fernandorevilla.es/iii/paginas-111-120/120-integrales-de-fresnel

P.S. 1 Although it is in Spanish, I think that one can follow the outline looking only at the formulas.

P.S. 2 There is a typo in the second line of the pdf.:

It should be $I_2=\displaystyle\int_0^{+\infty}\sin x^2\;dx$ instead of $I_2=\displaystyle\int_0^{+\infty}\cos x^2\;dx$

 

[alyafey22]

I see that you are using contour integration to solve the integral .

Do you have another method to solve it ?

 

[Fernando Revilla]

Do you have another method to solve it ?

I know another metod (Laplace transform), but it is not in my site. Have a look (for example) here:

http://www.mymathforum.com/viewtopic.php?f=22&t=20045