oldbesseli - Modified Bessel functions of the first kind (I sub alpha).
oldbesselj - Bessel functions of the first kind (J sub alpha).
oldbesselk - Modified Bessel functions of the second kind (K sub alpha).
oldbessely - Bessel functions of the second kind (Y sub alpha).
These functions are obsolete, use besseli , besselj , besselk , bessely instead. Note however that the semantics of these two sets of functions are different.
oldbesseli(alpha,x) computes modified Bessel functions of the first kind (I sub alpha), for real, non-negative order alpha and real non negative argument x . besseli(alpha,x,2) computes besseli(alpha,x).*exp(-x) .
oldbesselj(alpha,x) computes Bessel functions of the first kind (J sub alpha), for real, non-negative order alpha and real non negative argument x .
oldbesselk(alpha,x) computes modified Bessel functions of the second kind (K sub alpha), for real, non-negative order alpha and real non negative argument x . besselk(alpha,x,2) computes besselk(alpha,x).*exp(x) .
oldbessely(alpha,x) computes Bessel functions of the second kind (Y sub alpha), for real, non-negative order alpha and real non negative argument x .
alpha and x may be vectors. The output is m -by- n with m = size(x,'*') , n = size(alpha,'*') whose (i,j) entry is oldbessel?(alpha(j),x(i)) .
Y_alpha and J_alpha Bessel functions are 2 independant solutions of the Bessel 's differential equation :
2 2 2
x y" + x y' + (x - alpha ) y = 0 , alpha >= 0
K_alpha and I_alpha modified Bessel functions are 2 independant solutions of the modified Bessel 's differential equation :
2 2 2
x y" + x y' - (x + alpha ) y = 0 , alpha >= 0
// example #1: display some I Bessel functions
x = linspace(0.01,10,5000)';
y = oldbesseli(0:4,x);
ys = oldbesseli(0:4,x,2);
xbasc()
subplot(2,1,1)
plot2d(x,y, style=2:6, leg="I0@I1@I2@I3@I4", rect=[0,0,6,10])
xtitle("Some modified Bessel functions of the first kind")
subplot(2,1,2)
plot2d(x,ys, style=2:6, leg="I0s@I1s@I2s@I3s@I4s", rect=[0,0,6,1])
xtitle("Some modified scaled Bessel functions of the first kind")
// example #2 : display some J Bessel functions
x = linspace(0,40,5000)';
y = besselj(0:4,x);
xbasc()
plot2d(x,y, style=2:6, leg="J0@J1@J2@J3@J4")
xtitle("Some Bessel functions of the first kind")
// example #3 : 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);
xbasc()
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)")
// example #4: display some K Bessel functions
x = linspace(0.01,10,5000)';
y = besselk(0:4,x);
ys = besselk(0:4,x,2);
xbasc()
subplot(2,1,1)
plot2d(x,y, style=2:6, leg="K0@K1@K2@K3@K4", rect=[0,0,6,10])
xtitle("Some modified Bessel functions of the second kind")
subplot(2,1,2)
plot2d(x,ys, style=2:6, leg="K0s@K1s@K2s@K3s@K4s", rect=[0,0,6,10])
xtitle("Some modified scaled Bessel functions of the second kind")
// example #5: plot severals Y Bessel functions
x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded for x -> 0+
y = bessely(0:4,x);
xbasc()
plot2d(x,y, style=2:6, leg="Y0@Y1@Y2@Y3@Y4", rect=[0,-1.5,40,0.6])
xtitle("Some Bessel functions of the second kind")
W. J. Cody, L. Stoltz (code from Netlib (specfun))