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I would like to approximate the Fresnel S and Fresnel C with standard functions.

I've started with the $ S(x) $ function:

$$ approxS(x) = sgn(x) * \left ( sgn(x)* \left ( \frac{ \sin( \frac{x^2}{2} ) }{x} \right) + 0.5 \right ) $$

The result looks like this:

approx

This is the FresnelS function:

fresnelS

The difference ( $ approxS(x) - S(x) $ ):

difference

What do I wrong? How could I approximate it better?

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  • $\begingroup$ well to be fair to yourself, the error looks worse than it is due to the scale of your axes. $\endgroup$ – Chill2Macht Apr 30 '16 at 17:29
  • $\begingroup$ over what range of values of x? $\endgroup$ – jim Apr 30 '16 at 17:35
  • $\begingroup$ $ x \in \mathbb{R} $ $\endgroup$ – Iter Ator Apr 30 '16 at 17:42
  • $\begingroup$ You want one formula that is accurate everywhere? $\endgroup$ – jim Apr 30 '16 at 21:51
  • $\begingroup$ I am looking for a forula, which can approximate the Fresnel function everywhere as close, as possible $\endgroup$ – Iter Ator Apr 30 '16 at 21:54
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According to https://en.wikipedia.org/wiki/Fresnel_integral, for large $x$, $S(x) =\sqrt{\dfrac{\pi}{2}}\left(\dfrac{sign(x)}{2} -(1+O(x^{-4}))\left(\dfrac{\cos(x^2)}{x\sqrt{2\pi}} +\dfrac{\sin(x^2)}{x^3\sqrt{8\pi}} \right)\right) $ and $C(x) =\sqrt{\dfrac{\pi}{2}}\left(\dfrac{sign(x)}{2} +(1+O(x^{-4}))\left(\dfrac{\sin(x^2)}{x\sqrt{2\pi}} +\dfrac{\cos(x^2)}{x^3\sqrt{8\pi}} \right)\right) $.

You can get the direct power series (good for small $x$) by expanding the integral term-by-term.

It might be interesting to see how close these are.

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