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besseli

Modified Bessel functions of the first kind (Iα).

besselj

Bessel functions of the first kind (Jα).

besselk

Modified Bessel functions of the second kind (Kα).

bessely

Bessel functions of the second kind (Yα).

besselh

Bessel functions of the third kind (aka Hankel functions)

Syntax

y = besseli(alpha, x [,ice])
y = besselj(alpha, x [,ice])
y = besselk(alpha, x [,ice])
y = bessely(alpha, x [,ice])
y = besselh(alpha, x)
y = besselh(alpha, K, x [,ice])

Arguments

x

real or complex vector.

alpha

real vector

ice

integer flag, with default value 0

K

integer, with possible values 1 or 2, the Hankel function type.

Description

  • besseli(alpha,x) computes modified Bessel functions of the first kind (Iα), for real order alpha and argument x. besseli(alpha,x,1) computes besseli(alpha,x).*exp(-abs(real(x))).

  • besselj(alpha,x) computes Bessel functions of the fisrt kind (Jα), for real order alpha and argument x. besselj(alpha,x,1) computes besselj(alpha,x).*exp(-abs(imag(x))).

  • besselk(alpha,x) computes modified Bessel functions of the second kind (Kα), for real order alpha and argument x. besselk(alpha,x,1) computes besselk(alpha,x).*exp(x).

  • bessely(alpha,x) computes Bessel functions of the second kind (Yalpha), for real order alpha and argument x. bessely(alpha,x,1) computes bessely(alpha,x).*exp(-abs(imag(x))).

  • besselh(alpha [,K] ,x) computes Bessel functions of the third kind (Hankel function H1 or H2 depending on K), for real order alpha and argument x. If omitted K is supposed to be equal to 1. besselh(alpha,1,x,1) computes besselh(alpha,1,x).*exp(-%i*x) and besselh(alpha,2,x,1) computes besselh(alpha,2,x).*exp(%i*x)

Remarks

If alpha and x are arrays of the same size, the result y is also that size. If either input is a scalar, it is expanded to the other input's size. If one input is a row vector and the other is a column vector, the result y is a two-dimensional table of function values.

Yα and Jα Bessel functions are 2 independent solutions of the Bessel 's differential equation :

x^2.(d^2y/d^2x) + x.dy/dx + (x^2 - alpha^2).y = 0, alpha ≥ 0

Kα and Iα modified Bessel functions are 2 independent solutions of the modified Bessel 's differential equation :

x^2.(d^2y/d^2x) + x.dy/dx + (alpha^2 - x^2).y = 0, alpha ≥ 0

Hα1 and Hα2, the Hankel functions of first and second kind, are linear linear combinations of Bessel functions of the first and second kinds:

H^1_α(z) = J_α(z) + i \cdot Y_α(z) \n H^2_α(z) = J_α(z) - i \cdot Y_α(z)

Examples

// besselI functions
// -----------------
x = linspace(0.01,10,5000)';
clf
subplot(2,1,1)
plot2d(x,besseli(0:4,x), style=2:6)
legend('I'+string(0:4),2);
xtitle("Some modified Bessel functions of the first kind")
subplot(2,1,2)
plot2d(x,besseli(0:4,x,1), style=2:6)
legend('I'+string(0:4),1);
xtitle("Some modified scaled Bessel functions of the first kind")

// besselJ functions
// -----------------
clf
x = linspace(0,40,5000)';
plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
legend('I'+string(0:4),1);
xtitle("Some Bessel functions of the first kind")

// use the fact that J_(1/2)(x) = sqrt(2/(x pi)) sin(x)
// to compare the algorithm of besselj(0.5,x) with a more direct formula
   x = linspace(0.1,40,5000)';
   y1 = besselj(0.5, x);
   y2 = sqrt(2 ./(%pi*x)).*sin(x);
   er = abs((y1-y2)./y2);
   ind = find(er > 0 & y2 ~= 0);
   clf()
   subplot(2,1,1)
   plot2d(x,y1,style=2)
   xtitle("besselj(0.5,x)")
   subplot(2,1,2)
   plot2d(x(ind), er(ind), style=2, logflag="nl")
   xtitle("relative error between 2 formulae for besselj(0.5,x)")

// besselK functions
// =================
   x = linspace(0.01,10,5000)';
   clf()
   subplot(2,1,1)
   plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10])
   legend('K'+string(0:4),1);
   xtitle("Some modified Bessel functions of the second kind")
   subplot(2,1,2)
   plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10])
   legend('K'+string(0:4),1);
   xtitle("Some modified scaled Bessel functions of the second kind")

// besselY functions
// =================
   x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded  for x -> 0+
   clf()
   plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6])
   legend('Y'+string(0:4),4);
   xtitle("Some Bessel functions of the second kind")

// besselH functions
// =================
   x=-4:0.025:2; y=-1.5:0.025:1.5;
   [X,Y] = ndgrid(x,y);
   H = besselh(0,1,X+%i*Y);
   clf();f=gcf();
   f.color_map=jet(16);
   contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5],fpf=" ")
   legends(string(0.2:0.2:3.2),1:16,"ur")
   xtitle("Level curves of |H1(0,z)|")

Used Functions

The source codes can be found in SCI/modules/special_functions/src/fortran/slatec and SCI/modules/special_functions/src/fortran

Slatec : dbesi.f, zbesi.f, dbesj.f, zbesj.f, dbesk.f, zbesk.f, dbesy.f, zbesy.f, zbesh.f

Drivers to extend definition area (Serge Steer INRIA): dbesig.f, zbesig.f, dbesjg.f, zbesjg.f, dbeskg.f, zbeskg.f, dbesyg.f, zbesyg.f, zbeshg.f

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with contributors 		Last updated:
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