(maxima.info)Bessel Functions

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15.2 Bessel Functions
=====================

 -- Function: bessel_j (<v>, <z>)

     The Bessel function of the first kind of order v and argument z.

     'bessel_j' is defined as

                          inf
                          ====       k  - v - 2 k  v + 2 k
                          \     (- 1)  2          z
                           >    --------------------------
                          /        k! gamma(v + k + 1)
                          ====
                          k = 0

     although the infinite series is not used for computations.

 -- Function: bessel_y (<v>, <z>)

     The Bessel function of the second kind of order v and argument z.

     'bessel_y' is defined as
                        cos(%pi v) bessel_j(v, z) - bessel_j(-v, z)
                        -------------------------------------------
                                       sin(%pi v)

     when v is not an integer.  When v is an integer n, the limit as v
     approaches n is taken.

 -- Function: bessel_i (<v>, <z>)

     The modified Bessel function of the first kind of order v and
     argument z.

     'bessel_i' is defined as
                              inf
                              ====   - v - 2 k  v + 2 k
                              \     2          z
                               >    -------------------
                              /     k! gamma(v + k + 1)
                              ====
                              k = 0

     although the infinite series is not used for computations.

 -- Function: bessel_k (<v>, <z>)

     The modified Bessel function of the second kind of order v and
     argument z.

     'bessel_k' is defined as
                     %pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z))
                     -------------------------------------------------
                                            2

     when v is not an integer.  If v is an integer n, then the limit as
     v approaches n is taken.

 -- Function: hankel_1 (<v>, <z>)

     The Hankel function of the first kind of order v and argument z
     (A&S 9.1.3).  'hankel_1' is defined as

             bessel_j(v,z) + %i * bessel_y(v,z)

     Maxima evaluates 'hankel_1' numerically for a complex order v and
     complex argument z in float precision.  The numerical evaluation in
     bigfloat precision is not supported.

     When 'besselexpand' is 'true', 'hankel_1' is expanded in terms of
     elementary functions when the order v is half of an odd integer.
     See 'besselexpand'.

     Maxima knows the derivative of 'hankel_1' wrt the argument z.

     Examples:

     Numerical evaluation:

          (%i1) hankel_1(1,0.5);
          (%o1)        0.24226845767487 - 1.471472392670243 %i
          (%i2) hankel_1(1,0.5+%i);
          (%o2)       - 0.25582879948621 %i - 0.23957560188301

     Expansion of 'hankel_1' when 'besselexpand' is 'true':

          (%i1) hankel_1(1/2,z),besselexpand:true;
                         sqrt(2) sin(z) - sqrt(2) %i cos(z)
          (%o1)          ----------------------------------
                                 sqrt(%pi) sqrt(z)

     Derivative of 'hankel_1' wrt the argument z.  The derivative wrt
     the order v is not supported.  Maxima returns a noun form:

          (%i1) diff(hankel_1(v,z),z);
                       hankel_1(v - 1, z) - hankel_1(v + 1, z)
          (%o1)        ---------------------------------------
                                          2
          (%i2) diff(hankel_1(v,z),v);
                                 d
          (%o2)                  -- (hankel_1(v, z))
                                 dv

 -- Function: hankel_2 (<v>, <z>)

     The Hankel function of the second kind of order v and argument z
     (A&S 9.1.4).  'hankel_2' is defined as

             bessel_j(v,z) - %i * bessel_y(v,z)

     Maxima evaluates 'hankel_2' numerically for a complex order v and
     complex argument z in float precision.  The numerical evaluation in
     bigfloat precision is not supported.

     When 'besselexpand' is 'true', 'hankel_2' is expanded in terms of
     elementary functions when the order v is half of an odd integer.
     See 'besselexpand'.

     Maxima knows the derivative of 'hankel_2' wrt the argument z.

     For examples see 'hankel_1'.

 -- Option variable: besselexpand
     Default value: 'false'

     Controls expansion of the Bessel functions when the order is half
     of an odd integer.  In this case, the Bessel functions can be
     expanded in terms of other elementary functions.  When
     'besselexpand' is 'true', the Bessel function is expanded.

          (%i1) besselexpand: false$
          (%i2) bessel_j (3/2, z);
                                              3
          (%o2)                      bessel_j(-, z)
                                              2
          (%i3) besselexpand: true$
          (%i4) bessel_j (3/2, z);
                                                  sin(z)   cos(z)
                                 sqrt(2) sqrt(z) (------ - ------)
                                                     2       z
                                                    z
          (%o4)                  ---------------------------------
                                             sqrt(%pi)

 -- Function: scaled_bessel_i (<v>, <z>)

     The scaled modified Bessel function of the first kind of order v
     and argument z.  That is, scaled_bessel_i(v,z) =
     exp(-abs(z))*bessel_i(v, z).  This function is particularly useful
     for calculating bessel_i for large z, which is large.  However,
     maxima does not otherwise know much about this function.  For
     symbolic work, it is probably preferable to work with the
     expression 'exp(-abs(z))*bessel_i(v, z)'.

 -- Function: scaled_bessel_i0 (<z>)

     Identical to 'scaled_bessel_i(0,z)'.

 -- Function: scaled_bessel_i1 (<z>)

     Identical to 'scaled_bessel_i(1,z)'.

 -- Function: %s [<u>,<v>] (<z>)
     Lommel's little s[u,v](z) function.  Probably Gradshteyn & Ryzhik
     8.570.1.