<< amell Fonctions spéciales beta >>
Aide de Scilab >> Fonctions spéciales > besseli

besseli

Modified Bessel functions of the first kind (I sub alpha).

besselj

Bessel functions of the first kind (J sub alpha).

besselk

Modified Bessel functions of the second kind (K sub alpha).

bessely

Bessel functions of the second kind (Y sub alpha).

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

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 resulty is a two-dimensional table of function values.

Y_alpha and J_alpha Bessel functions are 2 independent solutions of the Bessel 's differential equation :

K_alpha and I_alpha modified Bessel functions are 2 independent solutions of the modified Bessel 's differential equation :

H^1_alpha and H^2_alpha, the Hankel functions of first and second kind, are linear linear combinations of Bessel functions of the first and second kinds:

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();
   xset("fpf"," ")
   f.color_map=jetcolormap(16);
   contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5])
   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


Report an issue
<< amell Fonctions spéciales beta >>