pracma (version 1.1.6)

Fresnel Integrals: Fresnel Integrals

Description

(Normalized) Fresnel integrals S(x) and C(x)

Usage

fresnelS(x)
fresnelC(x)

Arguments

x

numeric vector.

Value

• Numeric vector of function values.

Details

The normalized Fresnel integrals are defined as $$S(x) = \int_0^x \sin(\pi/2 \, t^2) dt$$ $$C(x) = \int_0^x \cos(\pi/2 \, t^2) dt$$

This program computes the Fresnel integrals S(x) and C(x) using Fortran code by Zhang and Jin. The accuracy is almost up to Machine precision.

The functions are not (yet) truly vectorized, but use a call to `apply'. The underlying function .fresnel (not exported) computes single values of S(x) and C(x) at the same time.

References

Zhang, S., and J. Jin (1996). Computation of Special Functions. Wiley-Interscience.

See Also

gaussLegendre

Examples

##  Compute Fresnel integrals through Gauss-Legendre quadrature
f1 <- function(t) sin(0.5 * pi * t^2)
f2 <- function(t) cos(0.5 * pi * t^2)
for (x in seq(0.5, 2.5, by = 0.5)) {
    cgl <- gaussLegendre(51, 0, x)
    fs <- sum(cgl$w * f1(cgl$x))
    fc <- sum(cgl$w * f2(cgl$x))
    cat(formatC(c(x, fresnelS(x), fs, fresnelC(x), fc),
        digits = 8, width = 12, flag = "----"), "")
}

xs <- seq(0, 7.5, by = 0.025)
ys <- fresnelS(xs)
yc <- fresnelC(xs)

##  Function plot of the Fresnel integrals
plot(xs, ys, type = "l", col = "darkgreen",
    xlim = c(0, 8), ylim = c(0, 1),
    xlab = "", ylab = "", main = "Fresnel Integrals")
lines(xs, yc, col = "blue")
legend(6.25, 0.95, c("S(x)", "C(x)"), col = c("darkgreen", "blue"), lty = 1)
grid()

##  The Cornu (or Euler) spiral
plot(c(-1, 1), c(-1, 1), type = "n",
    xlab = "", ylab = "", main = "Cornu Spiral")
lines(ys, yc, col = "red")
lines(-ys, -yc, col = "red")
grid()