Can Fresnel integrals be expressed as a function of other functions/integrals?

Question

I need to use Fresnel integrals in C++. Is it possible to compute them with any of the new built-in functions of C++17? or do I have to implement my own solver?

Said otherwise: can Fresnel integrals be expressed as a function of some elliptic integrals (and others, see link above)?

Answer 1

The Fresnel integrals are related to the Erf function :

http://functions.wolfram.com/GammaBetaErf/FresnelS/27/01/

http://functions.wolfram.com/GammaBetaErf/FresnelC/27/01/

and to Hypergeometric1F2 function :

http://functions.wolfram.com/GammaBetaErf/FresnelS/26/01/01/

http://functions.wolfram.com/GammaBetaErf/FresnelC/26/01/01/

Answer 2

If you wish, you can approach the problems with a simple series expansion:

$${\displaystyle {\begin{aligned}S(x)&=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+3}}{(2n+1)!(4n+3)}}\\[6pt]C(x)&=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+1}}{(2n)!(4n+1)}}\end{aligned}}}$$

which follows from Taylor expanding the sines and cosines in the integral definitions.