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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 \(z\). See A&S eqn 9.1.10 and DLMF 10.2.E2.

  `bessel_j` is defined as

  $$ J_v(z) = \sum_{k=0}^{\infty }{{{\left(-1\right)^{k}\,\left(z\over 2\right)^{v+2\,k} }\over{k!\,\Gamma\left(v+k+1\right)}}} $$

  although the infinite series is not used for computations.

  When `besselexpand` is `true`, `bessel_j` is expanded in terms of elementary functions when the order \(v\) is half of an odd integer. See `besselexpand`.

  Categories: Bessel functions · Special functions ·

Function: bessel_y (v, z) ¶

  The Bessel function of the second kind of order \(v\) and argument \(z\). See A&S eqn 9.1.2 and DLMF 10.2.E3.

  `bessel_y` is defined as $$ Y_v(z) = {{\cos(\pi v)\, J_v(z) - J_{-v}(z)}\over{\sin{\pi v}}} $$

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

  When `besselexpand` is `true`, `bessel_y` is expanded in terms of elementary functions when the order \(v\) is half of an odd integer. See `besselexpand`.

  Categories: Bessel functions · Special functions ·

Function: bessel_i (v, z) ¶

  The modified Bessel function of the first kind of order \(v\) and argument \(z\). See A&S eqn 9.6.10 and DLMF 10.25.E2.

  `bessel_i` is defined as $$ I_v(z) = \sum_{k=0}^{\infty } {{1\over{k!\,\Gamma \left(v+k+1\right)}} {\left(z\over 2\right)^{v+2\,k}}} $$

  although the infinite series is not used for computations.

  When `besselexpand` is `true`, `bessel_i` is expanded in terms of elementary functions when the order \(v\) is half of an odd integer. See `besselexpand`.

  Categories: Bessel functions · Special functions ·

Function: bessel_k (v, z) ¶

  The modified Bessel function of the second kind of order \(v\) and argument \(z\). See A&S eqn 9.6.2 and DLMF 10.27.E4.

  `bessel_k` is defined as $$ K_v(z) = {{\pi\,\csc \left(\pi\,v\right)\,\left(I_{-v}(z)-I_{v}(z)\right)}\over{2}} $$

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

  When `besselexpand` is `true`, `bessel_k` is expanded in terms of elementary functions when the order \(v\) is half of an odd integer. See `besselexpand`.

  Categories: Bessel functions · Special functions ·

Function: hankel_1 (v, z) ¶

  The Hankel function of the first kind of order \(v\) and argument \(z\). See A&S eqn 9.1.3 and DLMF 10.4.E3.

  `hankel_1` is defined as

  $$ H^{(1)}_v(z) = J_v(z) + i Y_v(z) $$

  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

  Categories: Bessel functions · Special functions ·

Function: hankel_2 (v, z) ¶

  The Hankel function of the second kind of order \(v\) and argument \(z\). See A&S eqn 9.1.4 and DLMF 10.4.E3.

  `hankel_2` is defined as

  $$ H^{(2)}_v(z) = J_v(z) - i Y_v(z) $$

  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`.

  Categories: Bessel functions · Special functions ·

Option variable: besselexpand ¶

  Default value: `false`

  Controls expansion of the Bessel, Hankel and Struve functions when the order is half of an odd integer. In this case, the 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)
      (%i5) bessel_y(3/2,z);
                                              sin(z)    cos(z)
                          sqrt(2) sqrt(z) ((- ------) - ------)
                                                z          2
                                                          z
      (%o5)               -------------------------------------
                                        sqrt(%pi)
      (%i6) bessel_i(3/2,z);
                                            cosh(z)   sinh(z)
                           sqrt(2) sqrt(z) (------- - -------)
                                               z         2
                                                        z
      (%o6)                -----------------------------------
                                        sqrt(%pi)
      (%i7) bessel_k(3/2,z);
                                            1        - z
                                 sqrt(%pi) (- + 1) %e
                                            z
      (%o7)                      -----------------------
                                     sqrt(2) sqrt(z)

  Categories: Bessel functions · Simplification flags and variables · Special functions ·

Function: scaled_bessel_i (v, z) ¶

  The scaled modified Bessel function of the first kind of order \(v\) and argument \(z\). That is,

  $$ {\rm scaled\_bessel\_i}(v,z) = e^{-|z|} I_v(z). $$

  This function is particularly useful for calculating \(I_v(z)\) 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)`.

  Categories: Bessel functions ·

Function: scaled_bessel_i0 (z) ¶

  Identical to `scaled_bessel_i(0,z)`.

  Categories: Bessel functions · Special functions ·

Function: scaled_bessel_i1 (z) ¶

  Identical to `scaled_bessel_i(1,z)`.

  Categories: Bessel functions · Special functions ·

Function: %s [u,v] (z) ¶

  Lommel’s little \(s_{\mu,\nu}(z)\) function. (DLMF 11.9.E3)(G&R 8.570.1).

  This Lommel function is the particular solution of the inhomogeneous Bessel differential equation:

  $$ {d^2\over dz^2} + {1\over z}{dw\over dz} + \left(1-{\nu^2\over z^2}\right) w = z^{\mu-1} $$

  This can be defined by the series

  $$ s_{\mu,\nu}(z) = z^{\mu+1}\sum_{k=0}^{\infty} (-1)^k {z^{2k}\over a_{k+1}(\mu, \nu)} $$

  where

  $$ a_k(\mu,\nu) = \prod_{m=1}^k \left(\left(\mu + 2m-1\right)^2-\nu^2\right) = 4^k\left(\mu-\nu+1\over 2\right)_k \left(\mu+\nu+1\over 2\right)_k $$
  Categories: Bessel functions · Special functions ·

Function: slommel [u,v] (z) ¶

  Lommel’s big \(S_{\mu,\nu}(z)\) function. (DLMF 11.9.E5)(G&R 8.570.2).

  Lommels big S function is another particular solution of the inhomogeneous Bessel differential equation (see %s) defined for all values of \(\mu\) and \(\nu\) , where

  $$ \eqalign{ S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} & \Gamma\left({\mu\over 2} + {\nu\over 2} + {1\over 2}\right) \Gamma\left({\mu\over 2} - {\nu\over 2} + {1\over 2}\right) \cr & \times \left(\sin\left({(\mu-\nu)\pi\over 2}\right) J_{\nu}(z) - \cos\left({(\mu-\nu)\pi\over 2}\right) Y_{\nu}(z)\right) } $$

  When \(\mu\pm \nu\) ) is an odd negative integer, the limit must be used.

  Categories: Bessel functions · Special functions ·

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