Bessel functions of the third kind (aka Hankel functions) (original) (raw)

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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 orderalpha and argument x.besseli(alpha,x,1) computesbesseli(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) computesbesselj(alpha,x).*exp(-abs(imag(x))).
• besselk(alpha,x) computes modified Bessel functions of the second kind (K sub alpha), for real orderalpha and argument x.besselk(alpha,x,1) computesbesselk(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) computesbessely(alpha,x).*exp(-abs(imag(x))).
• besselh(alpha [,K] ,x) computes Bessel functions of the third kind (Hankel function H1 or H2 depending onK), 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) andbesselh(alpha,2,x,1) computesbesselh(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 resulty is a two-dimensional table of function values.

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 :

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

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(0,40,5000)'; clf() 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(0.1,40,5000)'; y1 = besselj(0.5, x); y2 = sqrt(2 ./(%pix)).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(0.01,10,5000)'; 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(0.1,40,5000)'; 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=-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)|")

Authors

• Amos, D. E., (SNLA)
• Daniel, S. L., (SNLA)
• Weston, M. K., (SNLA)

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