Scilab Function

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

### Calling Sequence

  `y = besselk(alpha,x)`
  `y = besselk(alpha,x,ice)`

### 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

`besselk(alpha,x)` computes modified Bessel functions of the second kind (K sub alpha), for real, non-negative order `alpha` and 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 `besselk(alpha(j),x(i))` .

K_alpha and I_alpha (see `besseli` ) 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

If `ice` is equal to 2 exponentialy scaled Bessel functions is computed (`K_alpha_scaled(x) = exp(x) K_alpha(x)`).

### Examples

    // example : 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")

### See Also

`besselj` , `besseli` , `bessely` ,

### Author

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