Maxima 5.37.3 Manual: 15. Special Functions

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# 15. Special Functions

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## 15.1 Introduction to Special Functions

Special function notation follows:

    bessel_j (index, expr)         Bessel function, 1st kind
    bessel_y (index, expr)         Bessel function, 2nd kind
    bessel_i (index, expr)         Modified Bessel function, 1st kind
    bessel_k (index, expr)         Modified Bessel function, 2nd kind

    hankel_1 (v,z)                 Hankel function of the 1st kind
    hankel_2 (v,z)                 Hankel function of the 2nd kind
    struve_h (v,z)                 Struve H function
    struve_l (v,z)                 Struve L function

    assoc_legendre_p[v,u] (z)      Legendre function of degree v and order u
    assoc_legendre_q[v,u] (z)      Legendre function, 2nd kind

    %f[p,q] ([], [], expr)         Generalized Hypergeometric function
    gamma (z)                      Gamma function
    gamma_greek (a,z)              Incomplete gamma function
    gamma_incomplete (a,z)         Tail of incomplete gamma function
    hypergeometric (l1, l2, z)     Hypergeometric function
    slommel
    %m[u,k] (z)                    Whittaker function, 1st kind
    %w[u,k] (z)                    Whittaker function, 2nd kind
    erfc (z)                       Complement of the erf function

    expintegral_e (v,z)            Exponential integral E
    expintegral_e1 (z)             Exponential integral E1
    expintegral_ei (z)             Exponential integral Ei
    expintegral_li (z)             Logarithmic integral Li
    expintegral_si (z)             Exponential integral Si
    expintegral_ci (z)             Exponential integral Ci
    expintegral_shi (z)            Exponential integral Shi
    expintegral_chi (z)            Exponential integral Chi

    kelliptic (z)                  Complete elliptic integral of the first
                                   kind (K)
    parabolic_cylinder_d (v,z)     Parabolic cylinder D function

@ref{Category: Bessel functions} · @ref{Category: Airy functions} · @ref{Category: Special 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.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

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.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

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.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

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.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

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

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

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`.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

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)

  @ref{Category: Bessel functions} · @ref{Category: Simplification flags and variables} @ref{Category: 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, 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)`.

  @ref{Category: Bessel functions}

Function: scaled_bessel_i0 (z)

  Identical to `scaled_bessel_i(0,z)`.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

Function: scaled_bessel_i1 (z)

  Identical to `scaled_bessel_i(1,z)`.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

Function: %s [u,v] (z)

  Lommel's little s[u,v](z) function. Probably Gradshteyn & Ryzhik 8.570.1.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

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## 15.3 Airy Functions

The Airy functions Ai(x) and Bi(x) are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 10.4.

`y = Ai(x)` and `y = Bi(x)` are two linearly independent solutions of the Airy differential equation `diff (y(x), x, 2) - x y(x) = 0`.

If the argument `x` is a real or complex floating point number, the numerical value of the function is returned.

Function: airy_ai (x)

  The Airy function Ai(x). (A&S 10.4.2)

  The derivative `diff (airy_ai(x), x)` is `airy_dai(x)`.

  See also `airy_bi`, `airy_dai`, `airy_dbi`.

  @ref{Category: Airy functions} · @ref{Category: Special functions}

Function: airy_dai (x)

  The derivative of the Airy function Ai `airy_ai(x)`.

  See `airy_ai`.

  @ref{Category: Airy functions} · @ref{Category: Special functions}

Function: airy_bi (x)

  The Airy function Bi(x). (A&S 10.4.3)

  The derivative `diff (airy_bi(x), x)` is `airy_dbi(x)`.

  See `airy_ai`, `airy_dbi`.

  @ref{Category: Airy functions} · @ref{Category: Special functions}

Function: airy_dbi (x)

  The derivative of the Airy Bi function `airy_bi(x)`.

  See `airy_ai` and `airy_bi`.

  @ref{Category: Airy functions} · @ref{Category: Special functions}

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## 15.4 Gamma and factorial Functions

The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.

Function: bffac (expr, n)

  Bigfloat version of the factorial (shifted gamma) function. The second argument is how many digits to retain and return, it's a good idea to request a couple of extra.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Numerical evaluation}

Function: bfpsi (n, z, fpprec)
Function: bfpsi0 (z, fpprec)

  `bfpsi` is the polygamma function of real argument z and integer order n. `bfpsi0` is the digamma function. `bfpsi0 (z, fpprec)` is equivalent to `bfpsi (0, z, fpprec)`.

  These functions return bigfloat values. fpprec is the bigfloat precision of the return value.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Numerical evaluation}

Function: cbffac (z, fpprec)

  Complex bigfloat factorial.

  `load ("bffac")` loads this function.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Complex variables} · @ref{Category: Numerical evaluation}

Function: gamma (z)

  The basic definition of the gamma function (A&S 6.1.1) is

                               inf
                              /
                              [     z - 1   - t
                   gamma(z) = I    t      %e    dt
                              ]
                              /
                               0

  Maxima simplifies `gamma` for positive integer and positive and negative rational numbers. For half integral values the result is a rational number times `sqrt(%pi)`. The simplification for integer values is controlled by `factlim`. For integers greater than `factlim` the numerical result of the factorial function, which is used to calculate `gamma`, will overflow. The simplification for rational numbers is controlled by `gammalim` to avoid internal overflow. See `factlim` and `gammalim`.

  For negative integers `gamma` is not defined.

  Maxima can evalute `gamma` numerically for real and complex values in float and bigfloat precision.

  `gamma` has mirror symmetry.

  When `gamma_expand` is `true`, Maxima expands `gamma` for arguments `z+n` and `z-n` where `n` is an integer.

  Maxima knows the derivate of `gamma`.

  Examples:

  Simplification for integer, half integral, and rational numbers:

      (%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
      (%o1)        [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
      (%i2) map('gamma,[1/2,3/2,5/2,7/2]);
                          sqrt(%pi)  3 sqrt(%pi)  15 sqrt(%pi)
      (%o2)   [sqrt(%pi), ---------, -----------, ------------]
                              2           4            8
      (%i3) map('gamma,[2/3,5/3,7/3]);
                                        2           1
                                2 gamma(-)  4 gamma(-)
                            2           3           3
      (%o3)          [gamma(-), ----------, ----------]
                            3       3           9

  Numerical evaluation for real and complex values:

      (%i4) map('gamma,[2.5,2.5b0]);
      (%o4)     [1.329340388179137, 1.3293403881791370205b0]
      (%i5) map('gamma,[1.0+%i,1.0b0+%i]);
      (%o5) [0.498015668118356 - .1549498283018107 %i,
                4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]

  `gamma` has mirror symmetry:

      (%i6) declare(z,complex)$
      (%i7) conjugate(gamma(z));
      (%o7)                  gamma(conjugate(z))

  Maxima expands `gamma(z+n)` and `gamma(z-n)`, when `gamma_expand` is `true`:

      (%i8) gamma_expand:true$

      (%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
                                     gamma(z)
      (%o9)             [z gamma(z), --------, z + 1]
                                      z - 1

  The deriviative of `gamma`:

      (%i10) diff(gamma(z),z);
      (%o10)                  psi (z) gamma(z)
                                 0

  See also `makegamma`.

  The Euler-Mascheroni constant is `%gamma`.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Special functions}

Function: log_gamma (z)

  The natural logarithm of the gamma function.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Special functions}

Function: gamma_greek (a, z)

  The lower incomplete gamma function (A&S 6.5.2):

                               z
                              /
                              [  a - 1   - t
          gamma_greek(a, z) = I t      %e    dt
                              ]
                              /
                               0

  See also `gamma_incomplete` (upper incomplete gamma function).

  @ref{Category: Gamma and factorial functions} · @ref{Category: Special functions}

Function: gamma_incomplete (a, z)

  The incomplete upper gamma function (A&S 6.5.3):

                                    inf
                                   /
                                   [     a - 1   - t
          gamma_incomplete(a, z) = I    t      %e    dt
                                   ]
                                   /
                                    z

  See also `gamma_expand` for controlling how `gamma_incomplete` is expressed in terms of elementary functions and `erfc`.

  Also see the related functions `gamma_incomplete_regularized` and `gamma_incomplete_generalized`.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Special functions}

Function: gamma_incomplete_regularized (a, z)

  The regularized incomplete upper gamma function (A&S 6.5.1):

      gamma_incomplete_regularized(a, z) =
                                              gamma_incomplete(a, z)
                                              ----------------------
                                                     gamma(a)

  See also `gamma_expand` for controlling how `gamma_incomplete` is expressed in terms of elementary functions and `erfc`.

  Also see `gamma_incomplete`.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Special functions}

Function: gamma_incomplete_generalized (a, z1, z1)

  The generalized incomplete gamma function.

      gamma_incomplete_generalized(a, z1, z2) =
                                                     z2
                                                    /
                                                    [    a - 1   - t
                                                    I   t      %e    dt
                                                    ]
                                                    /
                                                     z1

  Also see `gamma_incomplete` and `gamma_incomplete_regularized`.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Special functions}

Option variable: gamma_expand

  Default value: `false`

  `gamma_expand` controls expansion of `gamma_incomplete`. When `gamma_expand` is `true`, `gamma_incomplete(v,z)` is expanded in terms of `z`, `exp(z)`, and `erfc(z)` when possible.

      (%i1) gamma_incomplete(2,z);
      (%o1)                       gamma_incomplete(2, z)
      (%i2) gamma_expand:true;
      (%o2)                                true
      (%i3) gamma_incomplete(2,z);
                                                 - z
      (%o3)                            (z + 1) %e
      (%i4) gamma_incomplete(3/2,z);
                                    - z   sqrt(%pi) erfc(sqrt(z))
      (%o4)               sqrt(z) %e    + -----------------------
                                                     2

  @ref{Category: Gamma and factorial functions} · @ref{Category: Simplification flags and variables}

Option variable: gammalim

  Default value: 10000

  `gammalim` controls simplification of the gamma function for integral and rational number arguments. If the absolute value of the argument is not greater than `gammalim`, then simplification will occur. Note that the `factlim` switch controls simplification of the result of `gamma` of an integer argument as well.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Simplification flags and variables}

Function: makegamma (expr)

  Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.

  See also `makefact`.

  @ref{Category: Gamma and factorial functions}

Function: beta (a, b)

  The beta function is defined as `gamma(a) gamma(b)/gamma(a+b)` (A&S 6.2.1).

  Maxima simplifies the beta function for positive integers and rational numbers, which sum to an integer. When `beta_args_sum_to_integer` is `true`, Maxima simplifies also general expressions which sum to an integer.

  For a or b equal to zero the beta function is not defined.

  In general the beta function is not defined for negative integers as an argument. The exception is for a=-n, n a positive integer and b a positive integer with b<=n, it is possible to define an analytic continuation. Maxima gives for this case a result.

  When `beta_expand` is `true`, expressions like `beta(a+n,b)` and `beta(a-n,b)` or `beta(a,b+n)` and `beta(a,b-n)` with `n` an integer are simplified.

  Maxima can evaluate the beta function for real and complex values in float and bigfloat precision. For numerical evaluation Maxima uses `log_gamma`:

                 - log_gamma(b + a) + log_gamma(b) + log_gamma(a)
               %e

  Maxima knows that the beta function is symmetric and has mirror symmetry.

  Maxima knows the derivatives of the beta function with respect to a or b.

  To express the beta function as a ratio of gamma functions see `makegamma`.

  Examples:

  Simplification, when one of the arguments is an integer:

      (%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
                                     1   9      1
      (%o1)                         [--, -, ---------]
                                     12  4  a (a + 1)

  Simplification for two rational numbers as arguments which sum to an integer:

      (%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
                                3 %pi   2 %pi
      (%o2)                    [-----, -------, sqrt(2) %pi]
                                  8    sqrt(3)

  When setting `beta_args_sum_to_integer` to `true` more general expression are simplified, when the sum of the arguments is an integer:

      (%i3) beta_args_sum_to_integer:true$
      (%i4) beta(a+1,-a+2);
                                      %pi (a - 1) a
      (%o4)                         ------------------
                                    2 sin(%pi (2 - a))

  The possible results, when one of the arguments is a negative integer:

      (%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
                                          1  1    1
      (%o5)                            [- -, -, - -]
                                          3  6    3

  `beta(a+n,b)` or `beta(a-n)` with `n` an integer simplifies when `beta_expand` is `true`:

      (%i6) beta_expand:true$
      (%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
                          a beta(a, b)  beta(a, b) (b + a - 1)  a
      (%o7)              [------------, ----------------------, -]
                             b + a              a - 1           b

  Beta is not defined, when one of the arguments is zero:

      (%i7) beta(0,b);
      beta: expected nonzero arguments; found 0, b
       -- an error.  To debug this try debugmode(true);

  Numercial evaluation for real and complex arguments in float or bigfloat precision:

      (%i8) beta(2.5,2.3);
      (%o8) .08694748611299981

      (%i9) beta(2.5,1.4+%i);
      (%o9) 0.0640144950796695 - .1502078053286415 %i

      (%i10) beta(2.5b0,2.3b0);
      (%o10) 8.694748611299969b-2

      (%i11) beta(2.5b0,1.4b0+%i);
      (%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i

  Beta is symmetric and has mirror symmetry:

      (%i14) beta(a,b)-beta(b,a);
      (%o14)                                 0
      (%i15) declare(a,complex,b,complex)$
      (%i16) conjugate(beta(a,b));
      (%o16)                 beta(conjugate(a), conjugate(b))

  The derivative of the beta function wrt `a`:

      (%i17) diff(beta(a,b),a);
      (%o17)               - beta(a, b) (psi (b + a) - psi (a))
                                            0             0

  @ref{Category: Gamma and factorial functions}

Function: beta_incomplete (a, b, z)

  The basic definition of the incomplete beta function (A&S 6.6.1) is

              z
             /
             [         b - 1  a - 1
             I  (1 - t)      t      dt
             ]
             /
              0

  This definition is possible for realpart(a)>0 and realpart(b)>0 and abs(z)<1. For other values the incomplete beta function can be defined through a generalized hypergeometric function:

         gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z

  (See functions.wolfram.com for a complete definition of the incomplete beta function.)

  For negative integers a = -n and positive integers b=m with m<=n the incomplete beta function is defined through

                                  m - 1           k
                                  ====  (1 - m)  z
                            n - 1 \            k
                           z       >    -----------
                                  /     k! (n - k)
                                  ====
                                  k = 0

  Maxima uses this definition to simplify `beta_incomplete` for a a negative integer.

  For a a positive integer, `beta_incomplete` simplifies for any argument b and z and for b a positive integer for any argument a and z, with the exception of a a negative integer.

  For z=0 and realpart(a)>0, `beta_incomplete` has the specific value zero. For z=1 and realpart(b)>0, `beta_incomplete` simplifies to the beta function `beta(a,b)`.

  Maxima evaluates `beta_incomplete` numerically for real and complex values in float or bigfloat precision. For the numerical evaluation an expansion of the incomplete beta function in continued fractions is used.

  When the option variable `beta_expand` is `true`, Maxima expands expressions like `beta_incomplete(a+n,b,z)` and `beta_incomplete(a-n,b,z)` where n is a positive integer.

  Maxima knows the derivatives of `beta_incomplete` with respect to the variables a, b and z and the integral with respect to the variable z.

  Examples:

  Simplification for a a positive integer:

      (%i1) beta_incomplete(2,b,z);
                                             b
                                  1 - (1 - z)  (b z + 1)
      (%o1)                       ----------------------
                                        b (b + 1)

  Simplification for b a positive integer:

      (%i2) beta_incomplete(a,2,z);
                                                     a
                                    (a (1 - z) + 1) z
      (%o2)                         ------------------
                                        a (a + 1)

  Simplification for a and b a positive integer:

      (%i3) beta_incomplete(3,2,z);
                                                     3
                                    (3 (1 - z) + 1) z
      (%o3)                         ------------------
                                            12

  a is a negative integer and b<=(-a), Maxima simplifies:

      (%i4) beta_incomplete(-3,1,z);
                                             1
      (%o4)                              - ----
                                              3
                                           3 z

  For the specific values z=0 and z=1, Maxima simplifies:

      (%i5) assume(a>0,b>0)$
      (%i6) beta_incomplete(a,b,0);
      (%o6)                                 0
      (%i7) beta_incomplete(a,b,1);
      (%o7)                            beta(a, b)

  Numerical evaluation in float or bigfloat precision:

      (%i8) beta_incomplete(0.25,0.50,0.9);
      (%o8)                          4.594959440269333
      (%i9)  fpprec:25$
      (%i10) beta_incomplete(0.25,0.50,0.9b0);
      (%o10)                    4.594959440269324086971203b0

  For abs(z)>1 `beta_incomplete` returns a complex result:

      (%i11) beta_incomplete(0.25,0.50,1.7);
      (%o11)              5.244115108584249 - 1.45518047787844 %i

  Results for more general complex arguments:

      (%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
      (%o14)             2.726960675662536 - .3831175704269199 %i
      (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i);
      (%o15)             13.04649635168716 %i - 5.802067956270001
      (%i16)

  Expansion, when `beta_expand` is `true`:

      (%i23) beta_incomplete(a+1,b,z),beta_expand:true;
                                                             b  a
                         a beta_incomplete(a, b, z)   (1 - z)  z
      (%o23)             -------------------------- - -----------
                                   b + a                 b + a

      (%i24) beta_incomplete(a-1,b,z),beta_expand:true;
                                                                 b  a - 1
                 beta_incomplete(a, b, z) (- b - a + 1)   (1 - z)  z
      (%o24)     -------------------------------------- - ---------------
                                 1 - a                         1 - a

  Derivative and integral for `beta_incomplete`:

      (%i34) diff(beta_incomplete(a, b, z), z);
                                    b - 1  a - 1
      (%o34)                 (1 - z)      z
      (%i35) integrate(beta_incomplete(a, b, z), z);
                    b  a
             (1 - z)  z
      (%o35) ----------- + beta_incomplete(a, b, z) z
                b + a
                                             a beta_incomplete(a, b, z)
                                           - --------------------------
                                                       b + a
      (%i36) factor(diff(%, z));
      (%o36)              beta_incomplete(a, b, z)

  @ref{Category: Gamma and factorial functions}

Function: beta_incomplete_regularized (a, b, z)

  The regularized incomplete beta function (A&S 6.6.2), defined as

      beta_incomplete_regularized(a, b, z) =
                                            beta_incomplete(a, b, z)
                                            ------------------------
                                                   beta(a, b)

  As for `beta_incomplete` this definition is not complete. See functions.wolfram.com for a complete definition of `beta_incomplete_regularized`.

  `beta_incomplete_regularized` simplifies a or b a positive integer.

  For z=0 and realpart(a)>0, `beta_incomplete_regularized` has the specific value 0. For z=1 and realpart(b)>0, `beta_incomplete_regularized` simplifies to 1.

  Maxima can evaluate `beta_incomplete_regularized` for real and complex arguments in float and bigfloat precision.

  When `beta_expand` is `true`, Maxima expands `beta_incomplete_regularized` for arguments a+n or a-n, where n is an integer.

  Maxima knows the derivatives of `beta_incomplete_regularized` with respect to the variables a, b, and z and the integral with respect to the variable z.

  Examples:

  Simplification for a or b a positive integer:

      (%i1) beta_incomplete_regularized(2,b,z);
                                             b
      (%o1)                       1 - (1 - z)  (b z + 1)

      (%i2) beta_incomplete_regularized(a,2,z);
                                                     a
      (%o2)                         (a (1 - z) + 1) z

      (%i3) beta_incomplete_regularized(3,2,z);
                                                     3
      (%o3)                         (3 (1 - z) + 1) z

  For the specific values z=0 and z=1, Maxima simplifies:

      (%i4) assume(a>0,b>0)$
      (%i5) beta_incomplete_regularized(a,b,0);
      (%o5)                                 0
      (%i6) beta_incomplete_regularized(a,b,1);
      (%o6)                                 1

  Numerical evaluation for real and complex arguments in float and bigfloat precision:

      (%i7) beta_incomplete_regularized(0.12,0.43,0.9);
      (%o7)                         .9114011367359802
      (%i8) fpprec:32$
      (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0);
      (%o9)               9.1140113673598075519946998779975b-1
      (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
      (%o10)             .2865367499935403 %i - 0.122995963334684
      (%i11) fpprec:20$
      (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
      (%o12)      2.8653674999354036142b-1 %i - 1.2299596333468400163b-1

  Expansion, when `beta_expand` is `true`:

      (%i13) beta_incomplete_regularized(a+1,b,z);
                                                           b  a
                                                    (1 - z)  z
      (%o13) beta_incomplete_regularized(a, b, z) - ------------
                                                    a beta(a, b)
      (%i14) beta_incomplete_regularized(a-1,b,z);
      (%o14) beta_incomplete_regularized(a, b, z)
                                                           b  a - 1
                                                    (1 - z)  z
                                               - ----------------------
                                                 beta(a, b) (b + a - 1)

  The derivative and the integral wrt z:

      (%i15) diff(beta_incomplete_regularized(a,b,z),z);
                                    b - 1  a - 1
                             (1 - z)      z
      (%o15)                 -------------------
                                 beta(a, b)
      (%i16) integrate(beta_incomplete_regularized(a,b,z),z);
      (%o16) beta_incomplete_regularized(a, b, z) z
                                                                 b  a
                                                          (1 - z)  z
                a (beta_incomplete_regularized(a, b, z) - ------------)
                                                          a beta(a, b)
              - -------------------------------------------------------
                                         b + a

  @ref{Category: Gamma and factorial functions}

Function: beta_incomplete_generalized (a, b, z1, z2)

  The basic definition of the generalized incomplete beta function is

                   z2
                 /
                 [          b - 1  a - 1
                 I   (1 - t)      t      dt
                 ]
                 /
                  z1

  Maxima simplifies `beta_incomplete_regularized` for a and b a positive integer.

  For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies `beta_incomplete_generalized` to `beta_incomplete`. For realpart(b)>0 and z1=1 or z2=1, Maxima simplifies to an expression with `beta` and `beta_incomplete`.

  Maxima evaluates `beta_incomplete_regularized` for real and complex values in float and bigfloat precision.

  When `beta_expand` is `true`, Maxima expands `beta_incomplete_generalized` for a+n and a-n, n a positive integer.

  Maxima knows the derivative of `beta_incomplete_generalized` with respect to the variables a, b, z1, and z2 and the integrals with respect to the variables z1 and z2.

  Examples:

  Maxima simplifies `beta_incomplete_generalized` for a and b a positive integer:

      (%i1) beta_incomplete_generalized(2,b,z1,z2);
                         b                      b
                 (1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
      (%o1)      -------------------------------------------
                                  b (b + 1)
      (%i2) beta_incomplete_generalized(a,2,z1,z2);
                                    a                      a
                 (a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
      (%o2)      -------------------------------------------
                                  a (a + 1)
      (%i3) beta_incomplete_generalized(3,2,z1,z2);
                    2      2                       2      2
            (1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
      (%o3) -----------------------------------------------------------
                                        12

  Simplification for specific values z1=0, z2=0, z1=1, or z2=1:

      (%i4) assume(a > 0, b > 0)$
      (%i5) beta_incomplete_generalized(a,b,z1,0);
      (%o5)                    - beta_incomplete(a, b, z1)

      (%i6) beta_incomplete_generalized(a,b,0,z2);
      (%o6)                    - beta_incomplete(a, b, z2)

      (%i7) beta_incomplete_generalized(a,b,z1,1);
      (%o7)              beta(a, b) - beta_incomplete(a, b, z1)

      (%i8) beta_incomplete_generalized(a,b,1,z2);
      (%o8)              beta_incomplete(a, b, z2) - beta(a, b)

  Numerical evaluation for real arguments in float or bigfloat precision:

      (%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
      (%o9)                        .09638178086368676

      (%i10) fpprec:32$
      (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
      (%o10)               9.6381780863686935309170054689964b-2

  Numerical evaluation for complex arguments in float or bigfloat precision:

      (%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
      (%o11)           - .09625463003205376 %i - .003323847735353769
      (%i12) fpprec:20$
      (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
      (%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3

  Expansion for a+n or a-n, n a positive integer, when `beta_expand` is `true`:

      (%i14) beta_expand:true$

      (%i15) beta_incomplete_generalized(a+1,b,z1,z2);

                     b   a           b   a
             (1 - z1)  z1  - (1 - z2)  z2
      (%o15) -----------------------------
                         b + a
                            a beta_incomplete_generalized(a, b, z1, z2)
                          + -------------------------------------------
                                               b + a
      (%i16) beta_incomplete_generalized(a-1,b,z1,z2);

             beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
      (%o16) -------------------------------------------------------
                                      1 - a
                                          b   a - 1           b   a - 1
                                  (1 - z2)  z2      - (1 - z1)  z1
                                - -------------------------------------
                                                  1 - a

  Derivative wrt the variable z1 and integrals wrt z1 and z2:

      (%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
                                     b - 1   a - 1
      (%o17)               - (1 - z1)      z1
      (%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
      (%o18) beta_incomplete_generalized(a, b, z1, z2) z1
                                        + beta_incomplete(a + 1, b, z1)
      (%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
      (%o19) beta_incomplete_generalized(a, b, z1, z2) z2
                                        - beta_incomplete(a + 1, b, z2)

  @ref{Category: Gamma and factorial functions}

Option variable: beta_expand

  Default value: false

  When `beta_expand` is `true`, `beta(a,b)` and related functions are expanded for arguments like a+n or a-n, where n is an integer.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Simplification flags and variables}

Option variable: beta_args_sum_to_integer

  Default value: false

  When `beta_args_sum_to_integer` is `true`, Maxima simplifies `beta(a,b)`, when the arguments a and b sum to an integer.

  @ref{Category: Gamma and factorial functions} · @ref{Category: Simplification flags and variables}

Function: psi [n](x)

  The derivative of `log (gamma (x))` of order `n+1`. Thus, `psi[0](x)` is the first derivative, `psi[1](x)` is the second derivative, etc.

  Maxima does not know how, in general, to compute a numerical value of `psi`, but it can compute some exact values for rational args. Several variables control what range of rational args `psi` will return an exact value, if possible. See `maxpsiposint`, `maxpsinegint`, `maxpsifracnum`, and `maxpsifracdenom`. That is, x must lie between `maxpsinegint` and `maxpsiposint`. If the absolute value of the fractional part of x is rational and has a numerator less than `maxpsifracnum` and has a denominator less than `maxpsifracdenom`, `psi` will return an exact value.

  The function `bfpsi` in the `bffac` package can compute numerical values.

  @ref{Category: Gamma and factorial functions}

Option variable: maxpsiposint

  Default value: 20

  `maxpsiposint` is the largest positive value for which `psi[n](x)` will try to compute an exact value.

  @ref{Category: Gamma and factorial functions}

Option variable: maxpsinegint

  Default value: -10

  `maxpsinegint` is the most negative value for which `psi[n](x)` will try to compute an exact value. That is if x is less than `maxnegint`, `psi[n](x)` will not return simplified answer, even if it could.

  @ref{Category: Gamma and factorial functions}

Option variable: maxpsifracnum

  Default value: 6

  Let x be a rational number less than one of the form `p/q`. If `p` is greater than `maxpsifracnum`, then `psi[n](x)` will not try to return a simplified value.

  @ref{Category: Gamma and factorial functions}

Option variable: maxpsifracdenom

  Default value: 6

  Let x be a rational number less than one of the form `p/q`. If `q` is greater than `maxpsifracdenom`, then `psi[n](x)` will not try to return a simplified value.

  @ref{Category: Gamma and factorial functions}

Function: makefact (expr)

  Transforms instances of binomial, gamma, and beta functions in expr into factorials.

  See also `makegamma`.

  @ref{Category: Gamma and factorial functions}

Function: numfactor (expr)

  Returns the numerical factor multiplying the expression expr, which should be a single term.

  `content` returns the greatest common divisor (gcd) of all terms in a sum.

      (%i1) gamma (7/2);
                                15 sqrt(%pi)
      (%o1)                     ------------
                                     8
      (%i2) numfactor (%);
                                     15
      (%o2)                          --
                                     8

  @ref{Category: Expressions}

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## 15.5 Exponential Integrals

The Exponential Integral and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 5

Function: expintegral_e1 (z)

  The Exponential Integral E1(z) (A&S 5.1.1)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Function: expintegral_ei (z)

  The Exponential Integral Ei(z) (A&S 5.1.2)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Function: expintegral_li (z)

  The Exponential Integral Li(z) (A&S 5.1.3)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Function: expintegral_e (n,z)

  The Exponential Integral En(z) (A&S 5.1.4)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Function: expintegral_si (z)

  The Exponential Integral Si(z) (A&S 5.2.1)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Function: expintegral_ci (z)

  The Exponential Integral Ci(z) (A&S 5.2.2)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Function: expintegral_shi (z)

  The Exponential Integral Shi(z) (A&S 5.2.3)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Function: expintegral_chi (z)

  The Exponential Integral Chi(z) (A&S 5.2.4)

  @ref{Category: Exponential Integrals} · @ref{Category: Special functions}

Option variable: expintrep

  Default value: false

  Change the representation of the Exponential Integral to gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, expintegral_hyp

  @ref{Category: Exponential Integrals}

Option variable: expintexpand

  Default value: false

  Expand the Exponential Integral E[n](z) for half integral values in terms of Erfc or Erf and for positive integers in terms of Ei

  @ref{Category: Exponential Integrals}

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## 15.6 Error Function

The Error function and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 7

Function: erf (z)

  The Error Function erf(z) (A&S 7.1.1)

  See also flag `erfflag`.

  @ref{Category: Special functions}

Function: erfc (z)

  The Complementary Error Function erfc(z) (A&S 7.1.2)

  `erfc(z) = 1-erf(z)`

  @ref{Category: Special functions}

Function: erfi (z)

  The Imaginary Error Function.

  `erfi(z) = -%i*erf(%i*z)`

  @ref{Category: Special functions}

Function: erf_generalized (z1,z2)

  Generalized Error function Erf(z1,z2)

  @ref{Category: Special functions}

Function: fresnel_c (z)

  The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z). (A&S 7.3.1)

  The simplification fresnel_c(-x) = -fresnel_c(x) is applied when flag `trigsign` is true.

  The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied when flag `%iargs` is true.

  See flags `erf_representation` and `hypergeometric_representation`.

  @ref{Category: Special functions}

Function: fresnel_s (z)

  The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z). (A&S 7.3.2)

  The simplification fresnel_s(-x) = -fresnel_s(x) is applied when flag `trigsign` is true.

  The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied when flag `%iargs` is true.

  See flags `erf_representation` and `hypergeometric_representation`.

  @ref{Category: Special functions}

Option variable: erf_representation

  Default value: false

  When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are transformed to erf.

Option variable: hypergeometric_representation

  Default value: false

  Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c

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## 15.7 Struve Functions

The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 12.

Function: struve_h (v, z)

  The Struve Function H of order v and argument z. (A&S 12.1.1)

  @ref{Category: Special functions}

Function: struve_l (v, z)

  The Modified Struve Function L of order v and argument z. (A&S 12.2.1)

  @ref{Category: Special functions}

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## 15.8 Hypergeometric Functions

The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapters 13 and 15.

Maxima has very limited knowledge of these functions. They can be returned from function `hgfred`.

Function: %m [k,u] (z)

  Whittaker M function `M[k,u](z) = exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z)`. (A&S 13.1.32)

  @ref{Category: Special functions}

Function: %w [k,u] (z)

  Whittaker W function. (A&S 13.1.33)

  @ref{Category: Special functions}

Function: %f [p,q] ([a],[b],z)

  The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function, where `a` a list of length `p` and `b` a list of length `q`.

  @ref{Category: Bessel functions} · @ref{Category: Special functions}

Function: hypergeometric ([a1, ..., ap],[b1, ... ,bq], x)

  The hypergeometric function. Unlike Maxima's `%f` hypergeometric function, the function `hypergeometric` is a simplifying function; also, `hypergeometric` supports complex double and big floating point evaluation. For the Gauss hypergeometric function, that is p = 2 and q = 1, floating point evaluation outside the unit circle is supported, but in general, it is not supported.

  When the option variable `expand_hypergeometric` is true (default is false) and one of the arguments `a1` through `ap` is a negative integer (a polynomial case), `hypergeometric` returns an expanded polynomial.

  Examples:

      (%i1)  hypergeometric([],[],x);
      (%o1) %e^x

  Polynomial cases automatically expand when `expand_hypergeometric` is true:

      (%i2) hypergeometric([-3],[7],x);
      (%o2) hypergeometric([-3],[7],x)

      (%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true;
      (%o3) -x^3/504+3*x^2/56-3*x/7+1

  Both double float and big float evaluation is supported:

      (%i4) hypergeometric([5.1],[7.1 + %i],0.42);
      (%o4)       1.346250786375334 - 0.0559061414208204 %i
      (%i5) hypergeometric([5,6],[8], 5.7 - %i);
      (%o5)     .007375824009774946 - .001049813688578674 %i
      (%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
      (%o6) 7.37582400977494674506442010824b-3
                                - 1.04981368857867315858055393376b-3 %i

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## 15.9 Parabolic Cylinder Functions

The Parabolic Cylinder Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 19.

Maxima has very limited knowledge of these functions. They can be returned from function `hgfred`.

Function: parabolic_cylinder_d (v, z)

  The parabolic cylinder function `parabolic_cylinder_d(v,z)`. (A&S 19.3.1)

  @ref{Category: Special functions}

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## 15.10 Functions and Variables for Special Functions

Function: specint (exp(- s*t) * expr, t)

  Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions.

  The following special functions are handled by `specint`: incomplete gamma function, error functions (but not the error function `erfi`, it is easy to transform `erfi` e.g. to the error function `erf`), exponential integrals, bessel functions (including products of bessel functions), hankel functions, hermite and the laguerre polynomials.

  Furthermore, `specint` can handle the hypergeometric function `%f[p,q]([],[],z)`, the whittaker function of the first kind `%m[u,k](z)` and of the second kind `%w[u,k](z)`.

  The result may be in terms of special functions and can include unsimplified hypergeometric functions.

  When `laplace` fails to find a Laplace transform, `specint` is called. Because `laplace` knows more general rules for Laplace transforms, it is preferable to use `laplace` and not `specint`.

  `demo(hypgeo)` displays several examples of Laplace transforms computed by `specint`.

  Examples:

      (%i1) assume (p > 0, a > 0)$
      (%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
                                 sqrt(%pi)
      (%o2)                     ------------
                                       a 3/2
                                2 (p + -)
                                       4
      (%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
                    * exp(-p*t), t);
                                         - a/p
                               sqrt(a) %e
      (%o3)                    ---------------
                                      2
                                     p

  Examples for exponential integrals:

      (%i4) assume(s>0,a>0,s-a>0)$
      (%i5) ratsimp(specint(%e^(a*t)
                            *(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
                                   log(s)
      (%o5)                        ------
                                   s - a
      (%i6) logarc:true$

      (%i7) gamma_expand:true$

      radcan(specint((cos(t)*expintegral_si(t)
                           -sin(t)*expintegral_ci(t))*%e^(-s*t),t));
                                   log(s)
      (%o8)                        ------
                                    2
                                   s  + 1
      ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
                            -2*t*expintegral_ci(a*t))*%e^(-s*t),t));
                                     2    2
                                log(s  + a )
      (%o9)                     ------------
                                      2
                                     s

  Results when using the expansion of `gamma_incomplete` and when changing the representation to `expintegral_e1`:

      (%i10) assume(s>0)$
      (%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                                  1
                                  gamma_incomplete(-, k s)
                                                  2
      (%o11)                      ------------------------
                                     sqrt(%pi) sqrt(s)

      (%i12) gamma_expand:true$
      (%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                    erfc(sqrt(k) sqrt(s))
      (%o13)                        ---------------------
                                           sqrt(s)

      (%i14) expintrep:expintegral_e1$
      (%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
                                    a s
                              a s %e    expintegral_e1(a s) - 1
      (%o15)                - ---------------------------------
                                              a

  @ref{Category: Laplace transform}

Function: hgfred (a, b, t)

  Simplify the generalized hypergeometric function in terms of other, simpler, forms. a is a list of numerator parameters and b is a list of the denominator parameters.

  If `hgfred` cannot simplify the hypergeometric function, it returns an expression of the form `%f[p,q]([a], [b], x)` where p is the number of elements in a, and q is the number of elements in b. This is the usual `pFq` generalized hypergeometric function.

      (%i1) assume(not(equal(z,0)));
      (%o1)                          [notequal(z, 0)]
      (%i2) hgfred([v+1/2],[2*v+1],2*%i*z);

                           v/2                               %i z
                          4    bessel_j(v, z) gamma(v + 1) %e
      (%o2)               ---------------------------------------
                                             v
                                            z
      (%i3) hgfred([1,1],[2],z);

                                         log(1 - z)
      (%o3)                            - ----------
                                             z
      (%i4) hgfred([a,a+1/2],[3/2],z^2);

                                     1 - 2 a          1 - 2 a
                              (z + 1)        - (1 - z)
      (%o4)                   -------------------------------
                                       2 (1 - 2 a) z

  It can be beneficial to load orthopoly too as the following example shows. Note that L is the generalized Laguerre polynomial.

      (%i5) load(orthopoly)$
      (%i6) hgfred([-2],[a],z);

                                          (a - 1)
                                       2 L       (z)
                                          2
      (%o6)                            -------------
                                         a (a + 1)
      (%i7) ev(%);

                                        2
                                       z        2 z
      (%o7)                         --------- - --- + 1
                                    a (a + 1)    a

Function: lambert_w (z)

  The principal branch of Lambert's W function W(z), the solution of `z = W(z) * exp(W(z))`. (DLMF 4.13)

  @ref{Category: Special functions}

Function: generalized_lambert_w (k, z)

  The k-th branch of Lambert's W function W(z), the solution of `z = W(z) * exp(W(z))`. (DLMF 4.13)

  The principal branch, denoted Wp(z) in DLMF, is `lambert_w(z) = generalized_lambert_w(0,z)`.

  The other branch with real values, denoted Wm(z) in DLMF, is `generalized_lambert_w(-1,z)`.

  @ref{Category: Special functions}

Function: nzeta (z)

  The Plasma Dispersion Function `nzeta(z) = %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))`

  @ref{Category: Special functions}

Function: nzetar (z)

  Returns `realpart(nzeta(z))`.

  @ref{Category: Special functions}

Function: nzetai (z)

  Returns `imagpart(nzeta(z))`.

  @ref{Category: Special functions}

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