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:

 

image\fresnels01.gif

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");

 

image\frens.gif

 

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:

 

image\fresnels02.gif

 

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