# Bessel: Bessel Functions

## 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).

## Usage

 ```1 2 3 4``` ```besselI(x, nu, expon.scaled = FALSE) besselK(x, nu, expon.scaled = FALSE) besselJ(x, nu) besselY(x, nu) ```

## Arguments

 `x` numeric, ≥ 0. `nu` numeric; The order (maybe fractional and negative) of the corresponding Bessel function. `expon.scaled` logical; if `TRUE`, the results are exponentially scaled in order to avoid overflow (I(nu)) or underflow (K(nu)), respectively.

## Details

If `expon.scaled = TRUE`, exp(-x) I(x;nu), or exp(x) K(x;nu) are returned.

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`.

The current algorithms will give warnings about accuracy loss for large arguments. In some cases, these warnings are exaggerated, and the precision is perfect. For large `nu`, say in the order of millions, the current algorithms are rarely useful.

## Value

Numeric vector with the (scaled, if `expon.scaled = TRUE`) values of the corresponding Bessel function.

The length of the result is the maximum of the lengths of the parameters. All parameters are recycled to that length.

## Author(s)

Original Fortran code: W. J. Cody, Argonne National Laboratory
Translation to C and adaptation to R: Martin Maechler maechler@stat.math.ethz.ch.

## Source

The C code is a translation of Fortran routines from http://www.netlib.org/specfun/ribesl, ../rjbesl, etc. The four source code files for bessel[IJKY] each contain a paragraph “Acknowledgement” and “References”, a short summary of which is

besselI

based on (code) by David J. Sookne, see Sookne (1973)... Modifications... An earlier version was published in Cody (1983).

besselJ

as `besselI`

besselK

based on (code) by J. B. Campbell (1980)... Modifications...

besselY

draws heavily on Temme's Algol program for Y... and on Campbell's programs for Y_ν(x) .... ... heavily modified.

## References

Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York; Chapter 9: Bessel Functions of Integer Order.

In order of “Source” citation above:

Sockne, David J. (1973). Bessel Functions of Real Argument and Integer Order. Journal of Research of the National Bureau of Standards, 77B, 125–132.

Cody, William J. (1983). Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Transactions on Mathematical Software, 9(2), 242–245. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1145/357456.357462")}.

Campbell, J.B. (1980). On Temme's algorithm for the modified Bessel function of the third kind. ACM Transactions on Mathematical Software, 6(4), 581–586. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1145/355921.355928")}.

Campbell, J.B. (1979). Bessel functions J_nu(x) and Y_nu(x) of float order and float argument. Computer Physics Communications, 18, 133–142. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1016/0010-4655(79)90030-4")}.

Temme, Nico M. (1976). On the numerical evaluation of the ordinary Bessel function of the second kind. Journal of Computational Physics, 21, 343–350. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1016/0021-9991(76)90032-2")}.

Other special mathematical functions, such as `gamma`, Γ(x), and `beta`, B(x).
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82``` ```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)})) ```