Can You Compute Fresnel Integrals?

[unseenoi]

hi everyone can someone please help me out. This is not homework just getting ready for school
integral of sin(x^2)

 

[Cyosis]

Are you familiar with complex integration and the residue theorem?

 

[unseenoi]

no i am not

 

[Hootenanny]

no i am not

In that case, I would try a substitution.

 

[Cyosis]

Seeing as you say you're getting ready for school, are you still in high school? Also I just noticed that you didn't specify an interval to integrate over. Did you just make up this integral yourself? The reason I am asking this is that this function does not have a primitive function in terms of elementary functions.

 

[Hootenanny]

Seeing as you say you're getting ready for school, are you still in high school? Also I just noticed that you didn't specify an interval to integrate over. Did you just make up this integral yourself? The reason I am asking this is that this function does not have a primitive function in terms of elementary functions.

Well noted Cyosis, I presumed that by 'school' the OP meant grad school, which looking back now may have not been a wise assumption.

 

[g_edgar]

$\int \sin(x^2)\,dx$ is not elementary.

So "hints" like "try substitution" are not helpful.

 

[Hootenanny]

$\int \sin(x^2)\,dx$ is not elementary.

Indeed it is, as has already been pointed out.

So "hints" like "try substitution" are not helpful.

Really? How about substituting u=x², then expanding sin(u) about u=0 and performing term-wise integration? Does this not give the power-series definition of the Fresnel function S(x)?

 

[n!kofeyn]

Really? How about substituting u=x², then expanding sin(u) about u=0 and performing term-wise integration? Does this not give the power-series definition of the Fresnel function S(x)?

I think your substitution hint implied either u-substitution or integration by parts. There is no need to make a substitution to expand sin(x²) out into its power series.

 

[Barkan]

it is a fresnel intetgral

 

[Barkan]

There is no systematic way to compute Fresnel integrals as I know.
But there are several approximation methods

I found Peter L. Volegov's code in Matlab central. It uses a method proposed in the following : (ith an error of less then 1x10-9)

Klaus D. Mielenz, Computation of Fresnel Integrals. II
J. Res. Natl. Inst. Stand. Technol. 105, 589 (2000), pp 589-590

Or simply wiki Fresnel Integrals

 

[Barkan]

by the way it is suprising that nobody above heard of Fresnels.

 

[Count Iblis]

I think your substitution hint implied either u-substitution or integration by parts. There is no need to make a substitution to expand sin(x²) out into its power series.

It is useful if you want to derive an asymptotic expression for the case of the integral from zero to R for large R.

 

[Count Iblis]

If you're integrating from 0 to R, then for small R, you simply integrate the Taylor expansion term by term.

If R is large, you write the integral as an integral from zero to infinity minus the integral from R to infinity. The former integral is is number which you ca easily obtaoin using contour integration methods. The latter you compute by doing the substitution x^2 = u as suggested by Hootenanny, and then you do a relpeated partial integration, where you integrate the sin and differentiate the 1/sqrt(u). You iterate this, each time integrating the trigonometric term and differentiating the 1/u^(n+1/2). This then yields an asymptotic expansion with the last unevaluated integral as an error term.

 

[lurflurf]

There is no systematic way to compute Fresnel integrals as I know.
But there are several approximation methods

There is no systematic way to compute sine as I know.
But there are several approximation methods
thus it would be quite a surprize if fresnel integrals were easier