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

$$J_v(z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(\frac{z}{2}\right)}^{v+2k}}{k!\mathrm{\Gamma }(v+k+1)}$$

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

$$Y_v(z)=\frac{\cos (\pi v)J_v(z)-J_{-v}(z)}{\sin \pi v}$$

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

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

$$I_v(z)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!\mathrm{\Gamma }(v+k+1)}{\left(\frac{z}{2}\right)}^{v+2k}$$

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

$$K_v(z)=\frac{\pi \csc \left(\pi v\right)(I_{-v}(z)-I_v(z))}{2}$$

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

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

$$H_v^{(1)}(z)=J_v(z)+i{Y}_v(z)$$

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

$$H_v^{(2)}(z)=J_v(z)-i{Y}_v(z)$$

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,

$$\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{\_}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{\_}\mathrm{i}(v,z)=e^{-|z|}{I}_v(z).$$

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:

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

This can be defined by the series

$$s_{\mu ,\nu }(z)=z^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k\frac{z^{2k}}{a_{k+1}(\mu ,\nu )}$$

where

$$a_k(\mu ,\nu )=\prod _{m=1}^k({(\mu +2m-1)}^2-{\nu }^2)=4^k{\left(\frac{\mu -\nu +1}{2}\right)}_k{\left(\frac{\mu +\nu +1}{2}\right)}_k$$

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

$$\begin{array}{rl}{S}_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}& \mathrm{\Gamma }(\frac{\mu }{2}+\frac{\nu }{2}+\frac{1}{2})\mathrm{\Gamma }(\frac{\mu }{2}-\frac{\nu }{2}+\frac{1}{2})\\ & \times (\sin \left(\frac{(\mu -\nu )\pi }{2}\right)J_{\nu }(z)-\cos \left(\frac{(\mu -\nu )\pi }{2}\right)Y_{\nu }(z))\end{array}$$

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

Categories: Bessel functions · Special functions ·