# 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(πt2/2) 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.

GAMMALN

SININT

## References:

[1]   Abramowitz and Stegun

Handbook of Mathematical Functions (9th printing 1970)

US Gov. Printing Office

Section 7.3 p300