BesselFunctions           package:fOptions           R Documentation

_M_o_d_i_f_i_e_d _B_e_s_s_e_l _F_u_n_c_t_i_o_n_s

_D_e_s_c_r_i_p_t_i_o_n:

     A collection and description of special mathematical functions
     which compute the modified Bessel functions of integer order of
     the first and second kind as well  as their derivatives.      


     The functions are:

       'BesselI'   modified Bessel function of the 1st Kind,
       'BesselDI'  its derivative,
       'BesselK'   the modified Bessel function of the 3nd Kind,
       'BesselDK'  its derivative.

_U_s_a_g_e:

     BesselI(x, nu, expon.scaled = FALSE) 
     BesselK(x, nu, expon.scaled = FALSE)
     BesselDI(x, nu) 
     BesselDK(x, nu)

_A_r_g_u_m_e_n_t_s:

expon.scaled: a logical; if TRUE, the results are exponentially scaled. 

      nu: an integer value greater or equal to zero, the integer order
          of the modified Bessel function. 

       x: a positive numeric value or a vector of positive numerical 
          values.    

_V_a_l_u_e:

     The functions return the values of the selected special
     mathematical function.

_A_u_t_h_o_r(_s):

     Diethelm Wuertz for the Rmetrics R-port.

_R_e_f_e_r_e_n_c_e_s:

     Abramowitz M., Stegun I.A. (1972);  _Handbook of Mathematical
     Functions with Formulas, Graphs,  and Mathematical Tables_,  9th
     printing, New York, Dover Publishing.  

     Weisstein E.W. (2004); _MathWorld - A Wolfram Web Resource_,
     http://mathworld.wolfram.com

_E_x_a_m_p_l_e_s:

     ## SOURCE("fOptions.3D-BesselFunctions")

     ## Bessel I0 and K0 - 
        # Abramowitz-Stegun: Table 9.8, p. 416-422
        x = c(0.0, 0.01, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50)  
        data.frame(x, I = exp(-x)*BesselI(x, 0), K = exp(x)*BesselK(x, 0)) 
        # Compare with R's internal function:
        # data.frame(x, ratio = BesselI(x, 0) / besselI(x, 0))
        # data.frame(x, ratio = BesselK(x, 0) / besselK(x, 0))
        
     ## x = 0:
        c(BesselI(0, 0), BesselI(0, 1), BesselI(0, 2), BesselI(0, 5))
        # Compare with R's internal function:
        # c(besselI(0, 0), besselI(0, 1), besselI(0, 2), besselI(0, 5))
        c(BesselK(0, 0), BesselK(0, 1), BesselK(0, 2), BesselK(0, 5))
        # Compare with R's internal function:
        # c(besselK(0, 0), besselK(0, 1), besselK(0, 2), besselK(0, 5))
        
     ## Bessel I2 and K2 - 
        # Abramowitz-Stegun: Table 9.8, p. 416-422
        x = c(0.0, 0.01, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50)  
        data.frame(x, I = BesselI(x, 2)/x^2, K = BesselK(x, 2)*x^2) 
        # Compare with R's internal function:
        # data.frame(x, ratio = BesselI(x, 0) / besselI(x, 0))
        # data.frame(x, ratio = BesselK(x, 0) / besselK(x, 0))
        # data.frame(x, ratio = BesselI(x, 1) / besselI(x, 1))
        # data.frame(x, ratio = BesselK(x, 1) / besselK(x, 1))
        # data.frame(x, ratio = BesselI(x, 5) / besselI(x, 5))
        # data.frame(x, ratio = BesselK(x, 5) / besselK(x, 5))
        # data.frame(x, ratio = BesselI(x,50) / besselI(x,50))
        # data.frame(x, ratio = BesselK(x,50) / besselK(x,50))

