The R Project SVN – R – Diff – /trunk/src/library/base/man/Bessel.Rd

   \eqn{I_{\nu}}{I(nu)} and \eqn{K_{\nu}}{K(nu)}.
+}
 \arguments{
   \item{x}{numeric, \eqn{\ge 0}.}
-  \item{nu}{numeric; The \emph{order} (maybe fractional!) of the
+  \item{nu}{numeric; The \emph{order} (maybe fractional and negative) of
-    corresponding Bessel function.}
+    the corresponding Bessel function.}
   \item{expon.scaled}{logical; if \code{TRUE}, the results are
     exponentially scaled in order to avoid overflow
     (\eqn{I_{\nu}}{I(nu)}) or underflow (\eqn{K_{\nu}}{K(nu)}),
     respectively.}
   Campbell, J.B. (1980).
   On Temme's algorithm for the modified Bessel function of the third kind.
   \emph{ACM Transactions on Mathematical Software}, \bold{6}(4), 581--586.
   \doi{10.1145/355921.355928}.
   Campbell, J.B. (1979).
   Bessel functions J_nu(x) and Y_nu(x) of float order and float argument.
   \emph{Computer Physics Communications}, \bold{18}, 133--142.
   \doi{10.1016/0010-4655(79)90030-4}.
 plot(x, x, ylim = c(0, 6), ylab = "", type = "n",
      main = "Bessel Functions  I_nu(x)")
 for(nu in nus) lines(x, besselI(x, nu = nu), col = nu + 2)
 legend(0, 6, legend = paste("nu=", nus), col = nus + 2, lwd = 1)
-x <- seq(0, 40, length.out = 801); yl <- c(-.8, .8)
+x <- seq(0, 40, length.out = 801); yl <- c(-.5, 1)
 plot(x, x, ylim = yl, ylab = "", type = "n",
      main = "Bessel Functions  J_nu(x)")
+abline(h=0, v=0, lty=3)
 for(nu in nus) lines(x, besselJ(x, nu = nu), col = nu + 2)
-legend(32, -.18, legend = paste("nu=", nus), col = nus + 2, lwd = 1)
+legend("topright", legend = paste("nu=", nus), col = nus + 2, lwd = 1, bty="n")
-## Negative nu's :
+## Negative nu's --------------------------------------------------
 xx <- 2:7
 nu <- seq(-10, 9, length.out = 2001)
-op <- par(lab = c(16, 5, 7))
+## --- I() --- --- --- ---
 matplot(nu, t(outer(xx, nu, besselI)), type = "l", ylim = c(-50, 200),
         main = expression(paste("Bessel ", I[nu](x), " for fixed ", x,
                                 ",  as ", f(nu))),
         xlab = expression(nu))
 abline(v = 0, col = "light gray", lty = 3)
-legend(5, 200, legend = paste("x=", xx), col=seq(xx), lty=seq(xx))
+legend(5, 200, legend = paste("x=", xx), col=seq(xx), lty=1:5)
-par(op)
-x0 <- 2^(-20:10)
+## --- J() --- --- --- ---
+bJ <- t(outer(xx, nu, besselJ))
-plot(x0, x0^-8, log = "xy", ylab = "", type = "n",
+matplot(nu, bJ, type = "l", ylim = c(-500, 200),
-     main = "Bessel Functions  J_nu(x)  near 0\n log - log  scale")
+        xlab = quote(nu), ylab = quote(J[nu](x)),
-for(nu in sort(c(nus, nus+0.5)))
+        main = expression(paste("Bessel ", J[nu](x), " for fixed ", x)))
-    lines(x0, besselJ(x0, nu = nu), col = nu + 2)
+abline(v = 0, col = "light gray", lty = 3)
-legend(3, 1e50, legend = paste("nu=", paste(nus, nus+0.5, sep=",")),
+legend("topright", legend = paste("x=", xx), col=seq(xx), lty=1:5)
-       col = nus + 2, lwd = 1)
+## ZOOM into right part:
+matplot(nu[nu > -2], bJ[nu > -2,], type = "l",
+        xlab = quote(nu), ylab = quote(J[nu](x)),
+        main = expression(paste("Bessel ", J[nu](x), " for fixed ", x)))
+abline(h=0, v = 0, col = "gray60", lty = 3)
+legend("topright", legend = paste("x=", xx), col=seq(xx), lty=1:5)
+##---------------  x --> 0  -----------------------------
+x0 <- 2^seq(-16, 5, length.out=256)
+plot(range(x0), c(1e-40, 1), log = "xy", xlab = "x", ylab = "", type = "n",
+     main = "Bessel Functions  J_nu(x)  near 0\n log - log  scale") ; axis(2, at=1)
+for(nu in sort(c(nus, nus+0.5)))
+    lines(x0, besselJ(x0, nu = nu), col = nu + 2, lty= 1+ (nu\%\%1 > 0))
+legend("right", legend = paste("nu=", paste(nus, nus+0.5, sep=", ")),
+       col = nus + 2, lwd = 1, bty="n")
+x0 <- 2^seq(-10, 8, length.out=256)
-plot(x0, x0^-8, log = "xy", ylab = "", type = "n",
+plot(range(x0), 10^c(-100, 80), log = "xy", xlab = "x", ylab = "", type = "n",
-     main = "Bessel Functions  K_nu(x)  near 0\n log - log  scale")
+     main = "Bessel Functions  K_nu(x)  near 0\n log - log  scale") ; axis(2, at=1)
 for(nu in sort(c(nus, nus+0.5)))
-    lines(x0, besselK(x0, nu = nu), col = nu + 2)
+    lines(x0, besselK(x0, nu = nu), col = nu + 2, lty= 1+ (nu\%\%1 > 0))
-legend(3, 1e50, legend = paste("nu=", paste(nus, nus + 0.5, sep = ",")),
+legend("topright", legend = paste("nu=", paste(nus, nus + 0.5, sep = ", ")),
-       col = nus + 2, lwd = 1)
+       col = nus + 2, lwd = 1, bty="n")
 x <- x[x > 0]
 plot(x, x, ylim = c(1e-18, 1e11), log = "y", ylab = "", type = "n",
-     main = "Bessel Functions  K_nu(x)")
+     main = "Bessel Functions  K_nu(x)"); axis(2, at=1)
 for(nu in nus) lines(x, besselK(x, nu = nu), col = nu + 2)
 legend(0, 1e-5, legend=paste("nu=", nus), col = nus + 2, lwd = 1)
 yl <- c(-1.6, .6)
 plot(x, x, ylim = yl, ylab = "", type = "n",

The R Project SVN R

(root)/trunk/src/library/base/man/Bessel.Rd – Rev 74265 → 75809