One of my Further Maths students has returned from his interview at Nottingham with a problem I am struggling with. He has asked me to integrate cos(x² )with respect to x. I anticipated no problems, but sadly have not been able to solve it

This seems like a little bit of a harsh question for an interview because...

You can't do that integral in closed form. By closed form I mean that you can't write down an expression for it in terms of any functions that you know. It is a type of integral called a Fresnel Integral and has important applications in the physics of diffraction and how to drive a motorcar round a corner quickly. Although I'm not going to explain why any of these are true - the A-level physics teacher should hopefully be able to expand on these cryptic statements!

Of course, if the question was to integrate cos(x² ) from 0 to infinity then it would be possible to do it. It is just the integral from 0 to y that is hard. To do this you just need to know how to integrate the function exp(ix² ). And then you take the real part to get the integral you want.

If you want some more information on Fresnel integrals then there is a good page here . Be sure to look at the section on the Cornu spiral too.



AlexB.

Is it possible to prove that the Fresnel integrals cannot be evaluated in closed form? By the way, another physics problem that needs Fresnel integrals is in the one-one section, in the Erratic Non-Linear oscillators topic.

Thanks,

Michael

Yes it is but I can't remember how... The best place to look may be in theoretical computer science papers - they do lots of stuff about this sort of thing.

AlexB.

Express $\cos (X)$ as (the first few terms of) an infinite series $(1-((X^2)/2!)+((X^4)/4!)-...)$, then substitute $X=x^2$. You can then integrate this infinite series $\mathrm{dx}$, in order to get an approximation for the area between the curve and the $x$-axis, i.e. an approximation for definite integrals (note that this only works over very small ranges, for instance $-{\pi }^2/4\le x\le {\pi }^2/4$, due to the oscillatory nature of the function. Unfortunately, it's not even periodic.