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

 

[blue_raver22]

Thank you for sharing this interesting unsolved question from MHF. The Fresnel integrals, defined as int_0^x sin(t^2) dt and int_0^x cos(t^2) dt, have been studied extensively in mathematical analysis and have various applications in physics and engineering. However, the exact evaluation of these integrals for all values of x is still an open question in mathematics.

In terms of a starting point for proving that int_0^inf sin(t^2) dt = sqrt(pi/8), I would recommend looking into various techniques for evaluating improper integrals, such as the Cauchy principal value or the method of contour integration. Additionally, exploring the properties of the Fresnel integrals and their relationship to other special functions, such as the error function, could provide valuable insights.

I hope this comment provides some helpful ideas for approaching this unsolved question. Good luck in your exploration of the Fresnel integrals!