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).

Calling Sequence

y = oldbesseli(alpha,x)

y = oldbesseli(alpha,x,ice)

y = oldbesselj(alpha,x)

y = oldbesselk(alpha,x)

y = oldbesselk(alpha,x,ice)

y = oldbessely(alpha,x)

Parameters

• 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

Description

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)) .

Remarks

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

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);
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(0:2,x);
ys = besselk(0:2,x,1);
xbasc()
subplot(2,1,1)
   plot2d(x,y, style=2:4, leg="K0@K1@K2", rect=[0,0,6,10])
   xtitle("Some modified Bessel functions of the second kind")
subplot(2,1,2)
   plot2d(x,ys, style=2:4, leg="K0s@K1s@K2s", 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(0:2,x);
xbasc()
plot2d(x,y, style=2:4, leg="Y0@Y1@Y2", rect=[0,-1.5,40,0.6])
xtitle("Some Bessel functions of the second kind")
 
  

Author

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