R语言base包 Bessel函数使用说明

返回R语言base包函数列表


功能\作用概述:

整数和分数阶贝塞尔函数,第一类和第二类,J(nu)和Y(nu),以及修正贝塞尔函数(第一类和第三类),I(nu)和K(nu)。


语法\用法:

besselI(x, nu, expon.scaled = FALSE)
besselK(x, nu, expon.scaled = FALSE)
besselJ(x, nu)
besselY(x, nu)


参数说明:

x : 数字,≥0。

nu : 数值;相应贝塞尔函数的阶数(可能是分数和负)。

expon.scaled : 逻辑的;如果为真,则结果按指数缩放,以分别避免溢出(I(nu))或下溢(K(nu))。


示例\实例:

require(graphics)

nus < - c(0:5, 10, 20)

x < - seq(0, 4, length.out = 501)
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(-.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("topright", legend = paste("nu=", nus), col = nus + 2, lwd = 1, bty="n")

## Negative nu's --------------------------------------------------
xx < - 2:7
nu < - seq(-10, 9, length.out = 2001)
## --- 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=1:5)

## --- J() --- --- --- ---
bJ < - t(outer(xx, nu, besselJ))
matplot(nu, bJ, type = "l", ylim = c(-500, 200),
xlab = quote(nu), ylab = quote(J[nu](x)),
main = expression(paste("Bessel ", J[nu](x), " for fixed ", x)))
abline(v = 0, col = "light gray", lty = 3)
legend("topright", legend = paste("x=", xx), col=seq(xx), lty=1:5)

## 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(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") ; axis(2, at=1)
for(nu in sort(c(nus, nus+0.5)))
lines(x0, besselK(x0, nu = nu), col = nu + 2, lty= 1+ (nu%%1 > 0))
legend("topright", legend = paste("nu=", paste(nus, nus + 0.5, sep = ", ")),
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)"); 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",
main = "Bessel Functions Y_nu(x)")
for(nu in nus){
xx < - x[x > .6*nu]
lines(xx, besselY(xx, nu=nu), col = nu+2)
}
legend(25, -.5, legend = paste("nu=", nus), col = nus+2, lwd = 1)

## negative nu in bessel_Y -- was bogus for a long time
curve(besselY(x, -0.1), 0, 10, ylim = c(-3,1), ylab = "")
for(nu in c(seq(-0.2, -2, by = -0.1)))
curve(besselY(x, nu), add = TRUE)
title(expression(besselY(x, nu) * " " *
{nu == list(-0.1, -0.2, ..., -2)}))