Bessel(base)
Bessel()所属R语言包：base
Bessel Functions
贝塞尔函数
译者：生物统计家园网 机器人LoveR
描述----------Description----------
Bessel Functions of integer and fractional order, of first and second kind, J(nu) and Y(nu), and Modified Bessel functions (of first and third kind), I(nu) and K(nu).
第一类和第二类，整数和分数阶Bessel函数J(nu)和Y(nu)，修正Bessel函数（第一和第三类），I(nu)和K(nu)。
用法----------Usage----------
besselI(x, nu, expon.scaled = FALSE)
besselK(x, nu, expon.scaled = FALSE)
besselJ(x, nu)
besselY(x, nu)
参数----------Arguments----------
参数：x
numeric, ≥ 0.
数字≥ 0。
参数：nu
numeric; The order (maybe fractional!) of the corresponding Bessel function.
数字;相应的Bessel函数的顺序（可能是小数）。
参数：expon.scaled
logical; if TRUE, the results are exponentially scaled in order to avoid overflow (I(nu)) or underflow (K(nu)), respectively.
逻辑;如果TRUE，结果被成倍放大，以避免溢出（I(nu)）或下溢（K(nu)），分别。
Details
详情----------Details----------
If expon.scaled = TRUE, exp(-x) I(x;nu), or exp(x) K(x;nu) are returned.
如果expon.scaled = TRUE，exp(-x) I(x;nu)或exp(x) K(x;nu)返回。
For nu < 0, formulae 9.1.2 and 9.6.2 from Abramowitz & Stegun are applied (which is probably suboptimal), except for besselK which is symmetric in nu.
nu < 0，从阿布拉莫维茨和Stegun公式9.1.2和9.6.2的应用（这可能是最理想的），除了besselK这是对称nu。
值----------Value----------
Numeric vector of the same length of x with the (scaled, if expon.scaled=TRUE) values of the corresponding Bessel function.
相同长度的数字矢量x（缩放，如果expon.scaled=TRUE）相应的Bessel函数值。
作者（S）----------Author(s)----------
Original Fortran code:
W. J. Cody, Argonne National Laboratory <br>
Translation to C and adaption to <font face="Courier New,Courier" color="#666666"><b>R</b></font>:
Martin Maechler <a href="mailto:maechler@stat.math.ethz.ch">maechler@stat.math.ethz.ch</a>.
源----------Source----------
The C code is a translation of Fortran routines from http://www.netlib.org/specfun/ribesl, ../rjbesl, etc.
C代码的Fortran例程从http://www.netlib.org/specfun/ribesl翻译，../rjbesl等
参考文献----------References----------
Handbook of Mathematical Functions. Dover, New York; Chapter 9: Bessel Functions of Integer Order.
参见----------See Also----------
Other special mathematical functions, such as gamma, Γ(x), and beta, B(x).
其他特殊的数学函数，如gamma，Γ(x)，beta，B(x)。
举例----------Examples----------
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(-.8, .8)
plot(x, x, ylim = yl, ylab = "", type = "n",
main = "Bessel Functions J_nu(x)")
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)
## Negative nu's :[＃负NU的：]
xx <- 2:7
nu <- seq(-10, 9, length.out = 2001)
op <- par(lab = c(16, 5, 7))
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))
par(op)
x0 <- 2^(-20:10)
plot(x0, x0^-8, log="xy", ylab="",type="n",
main = "Bessel Functions J_nu(x) near 0\n log - log scale")
for(nu in sort(c(nus, nus+.5)))
lines(x0, besselJ(x0, nu=nu), col = nu+2)
legend(3, 1e50, legend = paste("nu=", paste(nus, nus+.5, sep=",")),
col = nus + 2, lwd = 1)
plot(x0, x0^-8, log="xy", ylab="", type="n",
main = "Bessel Functions K_nu(x) near 0\n log - log scale")
for(nu in sort(c(nus, nus+.5)))
lines(x0, besselK(x0, nu=nu), col = nu+2)
legend(3, 1e50, legend = paste("nu=", paste(nus, nus+.5, sep=",")),
col = nus + 2, lwd = 1)
x <- x[x > 0]
plot(x, x, ylim=c(1e-18, 1e11), log = "y", ylab = "", type = "n",
main = "Bessel Functions K_nu(x)")
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[＃负bessel_Y NU - 是假的很长一段时间]
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)}))
转载请注明:出自 生物统计家园网(http://www.biostatistic.net)。
注：
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发表于 2012-2-16 19:56:19
