Modified Bessel functions of the first kind (I sub alpha).
Bessel functions of the first kind (Jα).
Modified Bessel functions of the second kind (Kα).
Bessel functions of the second kind (Yα).
Bessel functions of the third kind (aka Hankel functions)
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])
real or complex vector.
real vector
integer flag, with default value 0
integer, with possible values 1 or 2, the Hankel function type.
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)
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 :
Kα and Iα modified Bessel functions are 2 independent solutions of the modified Bessel 's differential equation :
Hα1 and Hα2, the Hankel functions of first and second kind, are linear linear combinations of Bessel functions of the first and second kinds:
// 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)|") | ![]() | ![]() |
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