Fresnel Integrals
(Normalized) Fresnel integrals S(x) and C(x)
fresnelS(x) fresnelC(x)
x |
numeric vector. |
The normalized Fresnel integrals are defined as
S(x) = \int_0^x \sin(π/2 \, t^2) dt
C(x) = \int_0^x \cos(π/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.
Numeric vector of function values.
Copyright (c) 1996 Zhang and Jin for the Fortran routines, converted to Matlab using the open source project ‘f2matlab’ by Ben Barrowes, posted to MatlabCentral in 2004, and then translated to R by Hans W. Borchers.
Zhang, S., and J. Jin (1996). Computation of Special Functions. Wiley-Interscience.
## 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 = " ----"), "\n")
}
## Not run:
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()
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.