FRESNELC

Purpose:

Evaluates the Fresnel Cosine Integral.

Syntax:

FRESNELC(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 C(x), the integration of cos(πt2/2) from 0 to x.

Example:

fresnelc(1)

 

returns 0.779893 the value of:

 

image\fresnelc01.gif

Example:

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

 

returns {0.099998, 0.199921, 0.779893, 0.488253}, the value of the integral with limits {0.1, 0.2, 1, 2}.

Example:

fresnelc(-5..0.01..5);

xlabel("x");ylabel("C(x)");label("Cosine Fresnel");

 

image\frencpic.gif

 

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

Remarks:

The Fresnel Cosine Integral, C(x), is defined as:

 

image\fresnelc02.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 FRESNELS to evaluate the sine form of the Fresnel integral.

 

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

See Also:

COSINT

ERF

FRESNELS

GAMMA

GAMMALN

References:

[1]   Abramowitz and Stegun

       Handbook of Mathematical Functions (9th printing 1970)

       US Gov. Printing Office

       Section 7.3 p300