# FRESNELS

## Purpose:

Evaluates the Fresnel Sine Integral.

## Syntax:

FRESNELS(x, xc)

x |
- |
A real or series, the integration limit. |

xc |
- |
Optional. A real, the cutoff limit that determines
the integration limit. Defaults to 1.8. |

## Returns:

A scalar or series, the value of S(x), the integration of sin(*πt*^{2}/2)
from 0 to *x*.

## Example:

fresnels(1)

returns 0.438259 the value of:

## Example:

fresnels({0.1, 0.2, 1, 2})

returns {0.000524, 0.004188, 0.438259,
0.343416}, the value of the integral with limits {0.1, 0.2, 1, 2}.

## Example:

fresnels(-5..0.01..5);

xlabel("x");ylabel("S(x)");label("Sine
Fresnel");

returns 1001 samples of the Fresnel Sine Integral with integration limits
from –5 to 5.

## Remarks:

The Fresnel Sine Integral, S(x), is
defined as:

For **abs(x) < xc**, a power
series about **x**
= 0 is used yielding an
accuracy better than 5e-16.

For **abs(x) > xc**, a minimax
rational approximation based on auxilliary functions described in [1]
is used yielding an accuracy better than 1e-9.

See FRESNELC to evaluate the cosine form
of the Fresnel integral.

FRESNELS was developed from an algorithm by J. N. McElwaine.

## See Also:

ERF

FRESNELC

GAMMA

GAMMALN

SININT

## References:

[1] Abramowitz
and Stegun

*Handbook of Mathematical Functions*
(9th printing 1970)

US
Gov. Printing Office

Section
7.3 p300