# besseli

Modified Bessel functions of the first kind (I sub alpha).

# besselj

Bessel functions of the first kind (J sub alpha).

# besselk

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

# bessely

Bessel functions of the second kind (Y sub alpha).

# besselh

Bessel functions of the third kind (aka Hankel functions)

## Calling Sequence

    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])

## Arguments

x

  real or complex vector.

alpha

  real vector

ice

  integer flag, with default value 0

K

  integer, with possible values 1 or 2, the Hankel function type.

## Description

- `besseli(alpha,x)` computes modified Bessel functions of the first kind (I sub alpha), 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 first kind (J sub alpha), 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 (K sub alpha), 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 (Y sub alpha), 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)`

## Remarks

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.

Y_alpha and J_alpha Bessel functions are 2 independent 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 :

H^1_alpha and H^2_alpha, the Hankel functions of first and second kind, are linear linear combinations of Bessel functions of the first and second kinds:

## Examples

    //  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")

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")

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)")

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")

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")

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)|")

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)|")

## Used Functions

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