パラメータ

x

real vector with non negative entries

alpha

real vector with non negative entries regularly spaced with increment equal to one alpha=alpha0+(n1:n2)

ice

integer flag, with default value 1

説明

これらの関数は古い関数であり, besseli, besselj, besselk, bessely を代わりに使用してください. しかし,これらの2組の関数の構文は異なっていることに注意してください.

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)).

Remarks

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

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

Examples

// example #1: display some I Bessel functions
x = linspace(0.01,10,5000)';
y = oldbesseli(0:4,x);
ys = oldbesseli(0:4,x,2);
clf()
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);
clf()
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);
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)") 

// example #4: display some K Bessel functions
x = linspace(0.01,10,5000)';
y = besselk(0:4,x);
ys = besselk(0:4,x,1);
clf()
subplot(2,1,1)
plot2d(x,y, style=0:4, 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=0:4, 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);
clf()
plot2d(x,y, style=0:4, leg="Y0@Y1@Y2@Y3@Y4", rect=[0,-1.5,40,0.6])
xtitle("Some Bessel functions of the second kind")

Authors

W. J. Cody, L. Stoltz (code from Netlib (specfun))