Evaluates the Fresnel Sine Integral.
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. |
A scalar or series, the value of S(x), the integration of
fresnels(1)
returns 0.438259 the value of:
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}.
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.
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.
[1] Abramowitz and Stegun
Handbook of Mathematical Functions (9th printing 1970)
US Gov. Printing Office
Section 7.3 p300