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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.

(%i1) bffac(1/2,16);
(%o1)                        8.862269254527584b-1
(%i2) (1/2)!,numer;
(%o2)                          0.886226925452758
(%i3) bffac(1/2,32);
(%o3)                 8.862269254527580136490837416707b-1
Function: bfpsi (n, z, fpprec)
Function: bfpsi0 (z, fpprec)

bfpsi is the polygamma function of real argument z and integer order n. See psi for further information. 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.

(%i1) bfpsi0(1/3, 15);
(%o1)                        - 3.13203378002081b0
(%i2) bfpsi0(1/3, 32);
(%o2)                - 3.1320337800208063229964190742873b0
(%i3) bfpsi(0,1/3,32);
(%o3)                - 3.1320337800208063229964190742873b0
(%i4) psi[0](1/3);
                          3 log(3)       %pi
(%o4)                  (- --------) - --------- - %gamma
                             2        2 sqrt(3)
(%i5) float(%);
(%o5)                         - 3.132033780020806
Function: cbffac (z, fpprec)

Complex bigfloat factorial.

load ("bffac") loads this function.

(%i1) cbffac(1+%i,16);
(%o1)           3.430658398165453b-1 %i + 6.529654964201666b-1
(%i2) (1+%i)!,numer;
(%o2)             0.3430658398165453 %i + 0.6529654964201667
Function: gamma (z)

The basic definition of the gamma function (DLMF 5.2.E1 and A&S eqn 6.1.1) is

\[\Gamma\left(z\right)=\int_{0}^{\infty }{t^{z-1}\,e^ {- t }\;dt} \]

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 evaluate 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 derivative 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 derivative of gamma:

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

See also makegamma.

The Euler-Mascheroni constant is %gamma.

Function: log_gamma (z)

The natural logarithm of the gamma function.

(%i1) gamma(6);
(%o1)                                 120
(%i2) log_gamma(6);
(%o2)                              log(120)
(%i3) log_gamma(0.5);
(%o3)                         0.5723649429247004
Function: gamma_incomplete_lower (a, z)

The lower incomplete gamma function (DLMF 8.2.E1 and A&S eqn 6.5.2):

\[\gamma\left(a , z\right)=\int_{0}^{z}{t^{a-1}\,e^ {- t }\;dt} \]

See also gamma_incomplete (upper incomplete gamma function).

Function: gamma_incomplete (a, z)

The incomplete upper gamma function (DLMF 8.2.E2 and A&S eqn 6.5.3):

\[\Gamma\left(a , z\right)=\int_{z}^{\infty }{t^{a-1}\,e^ {- t }\;dt} \]

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.

Function: gamma_incomplete_regularized (a, z)

The regularized incomplete upper gamma function (DLMF 8.2.E4):

\[Q\left(a , z\right)={{\Gamma\left(a , z\right)}\over{\Gamma\left(a\right)}} \]

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

Also see gamma_incomplete.

Function: gamma_incomplete_generalized (a, z1, z1)

The generalized incomplete gamma function.

\[\Gamma\left(a , z_{1}, z_{2}\right)=\int_{z_{1}}^{z_{2}}{t^{a-1}\,e^ {- t }\;dt} \]

Also see gamma_incomplete and gamma_incomplete_regularized.

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 gamma_incomplete or erfc 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
(%i5) gamma_incomplete(4/3,z);
                                                    1
                                   gamma_incomplete(-, z)
                       1/3   - z                    3
(%o5)                 z    %e    + ----------------------
                                             3
(%i6) gamma_incomplete(a+2,z);
             a               - z
(%o6)       z  (z + a + 1) %e    + a (a + 1) gamma_incomplete(a, z)
(%i7) gamma_incomplete(a-2, z);
        gamma_incomplete(a, z)    a - 2         z            1      - z
(%o7)   ---------------------- - z      (--------------- + -----) %e
           (1 - a) (2 - a)               (a - 2) (a - 1)   a - 2

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.

Function: makegamma (expr)

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

See also makefact.

(%i1) makegamma(binomial(n,k));
                                 gamma(n + 1)
(%o1)                    -----------------------------
                         gamma(k + 1) gamma(n - k + 1)
(%i2) makegamma(x!);
(%o2)                            gamma(x + 1)
(%i3) makegamma(beta(a,b));
                               gamma(a) gamma(b)
(%o3)                          -----------------
                                 gamma(b + a)
Function: beta (a, b)

The beta function is defined as

\[{\rm B}(a, b) = {{\Gamma(a) \Gamma(b)}\over{\Gamma(a+b)}} \]

(DLMF 5.12.E1 and A&S eqn 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,b) 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);

Numerical 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
Function: beta_incomplete (a, b, z)

The basic definition of the incomplete beta function (DLMF 8.17.E1 and A&S eqn 6.6.1) is

\[{\rm B}_z(a,b) = \int_0^z t^{a-1}(1-t)^{b-1}\; dt \]

This definition is possible for \({\rm Re}(a) > 0\) and \({\rm Re}(b) > 0\) and \(|z| < 1\) . 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 https://functions.wolfram.com/GammaBetaErf/Beta3/ for a complete definition of the incomplete beta function.)

For negative integers \(a = -n\) and positive integers \(b=m\) with \(m \le n\) the incomplete beta function is defined through

\[z^{n-1}\sum_{k=0}^{m-1} {{(1-m)_k z^k} \over {k! (n-k)}} \]

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 \({\rm Re}(a) > 0\) , beta_incomplete has the specific value zero. For \(z=1\) and \({\rm Re}(b) > 0\) , 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)
Function: beta_incomplete_regularized (a, b, z)

The regularized incomplete beta function (DLMF 8.17.E2 and A&S eqn 6.6.2), defined as

\[I_z(a,b) = {{\rm B}_z(a,b)\over {\rm B}(a,b)} \]

As for beta_incomplete this definition is not complete. See https://functions.wolfram.com/GammaBetaErf/BetaRegularized/ for a complete definition of beta_incomplete_regularized.

beta_incomplete_regularized simplifies a or b a positive integer.

For \(z=0\) and \({\rm Re}(a)>0\) , beta_incomplete_regularized has the specific value 0. For \(z=1\) and \({\rm Re}(b) > 0\) , 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
Function: beta_incomplete_generalized (a, b, z1, z2)

The basic definition of the generalized incomplete beta function is

\[\int_{z_1}^{z_2} t^{a-1}(1-t)^{b-1}\; dt \]

Maxima simplifies beta_incomplete_regularized for a and b a positive integer.

For \({\rm Re}(a) > 0\) and \(z_1 = 0\) or \(z_2 = 0\) , Maxima simplifies beta_incomplete_generalized to beta_incomplete. For \({\rm Re}(b) > 0\) and \(z_1 = 1\) or \(z_2 = 1\) , 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)
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.

See beta for examples.

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.

See beta for examples.

Function: psi [n](x)

psi[n](x) is the polygamma function (DLMF 5.2E2, DLMF 5.15, A&S eqn 6.3.1 and A&S eqn 6.4.1) defined by

\[\psi^{(n)}(x) = {d^{n+1}\over{dx^{n+1}}} \log\Gamma(x) \]

Thus, psi[0](x) is the first derivative, psi[1](x) is the second derivative, etc.

Maxima can compute some exact values for rational args as well for float and bfloat args. Several variables control what range of rational args \(\psi^{(n)}(x)\) ) 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, \(\psi^{(0)}(x)\) will return an exact value.

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

(%i1) psi[0](.25);
(%o1)                        - 4.227453533376265
(%i2) psi[0](1/4);
                                        %pi
(%o2)                    (- 3 log(2)) - --- - %gamma
                                         2
(%i3) float(%);
(%o3)                        - 4.227453533376265
(%i4) psi[2](0.75);
(%o4)                        - 5.30263321633764
(%i5) psi[2](3/4);
                                   1         3
(%o5)                         psi (-) + 4 %pi
                                 2 4
(%i6) float(%);
(%o6)                        - 5.30263321633764
Option variable: maxpsiposint

Default value: 20

maxpsiposint is the largest positive integer value for which \(\psi^{(n)}(m)\) gives an exact value for rational \(x\).

(%i1) psi[0](20);
                             275295799
(%o1)                        --------- - %gamma
                             77597520
(%i2) psi[0](21);
(%o2)                             psi (21)
                                     0
(%i3) psi[2](20);
                      1683118856778495358491487
(%o3)              2 (------------------------- - zeta(3))
                      1401731326612193601024000
(%i4) psi[2](21);
(%o4)                            psi (21)
                                      2

Option variable: maxpsinegint

Default value: -10

maxpsinegint is the most negative value for which \(\psi^{(0)}(x)\) will try to compute an exact value for rational \(x\). That is if \(x\) is less than maxpsinegint, \(\psi^{(n)}(x)\) will not return simplified answer, even if it could.

(%i1) psi[0](-100/9);
                                        100
(%o1)                            psi (- ---)
                                    0    9
(%i2) psi[0](-100/11);
                         100 %pi         1     5231385863539
(%o2)            %pi cot(-------) + psi (--) + -------------
                           11          0 11    381905105400

(%i3) psi[2](-100/9);
                                        100
(%o3)                            psi (- ---)
                                    2    9
(%i4) psi[2](-100/11);
           3     100 %pi     2 100 %pi         1
(%o4) 2 %pi  cot(-------) csc (-------) + psi (--)
                   11            11          2 11
                                         74191313259470963498957651385614962459
                                       + --------------------------------------
                                          27850718060013605318710152732000000
Option variable: maxpsifracnum

Default value: 6

Let \(x\) be a rational number of the form \(p/q\). If \(p\) is greater than maxpsifracnum, then \(\psi^{(0)}(x)\) will not try to return a simplified value.

(%i1) psi[0](3/4);
                                        %pi
(%o1)                    (- 3 log(2)) + --- - %gamma
                                         2
(%i2) psi[2](3/4);
                                   1         3
(%o2)                         psi (-) + 4 %pi
                                 2 4
(%i3) maxpsifracnum:2;
(%o3)                                 2
(%i4) psi[0](3/4);
                                        3
(%o4)                              psi (-)
                                      0 4
(%i5) psi[2](3/4);
                                   1         3
(%o5)                         psi (-) + 4 %pi
                                 2 4
Option variable: maxpsifracdenom

Default value: 6

Let \(x\) be a rational number of the form \(p/q\). If \(q\) is greater than maxpsifracdenom, then \(\psi^{(0)}(x)\) will not try to return a simplified value.

(%i1) psi[0](3/4);
                                        %pi
(%o1)                    (- 3 log(2)) + --- - %gamma
                                         2
(%i2) psi[2](3/4);
                                   1         3
(%o2)                         psi (-) + 4 %pi
                                 2 4
(%i3) maxpsifracdenom:2;
(%o3)                                 2
(%i4) psi[0](3/4);
                                        3
(%o4)                              psi (-)
                                      0 4
(%i5) psi[2](3/4);
                                   1         3
(%o5)                         psi (-) + 4 %pi
                                 2 4
Function: makefact (expr)

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

See also makegamma.

(%i1) makefact(binomial(n,k));
                                      n!
(%o1)                             -----------
                                  k! (n - k)!
(%i2) makefact(gamma(x));
(%o2)                              (x - 1)!
(%i3) makefact(beta(a,b));
                               (a - 1)! (b - 1)!
(%o3)                          -----------------
                                 (b + a - 1)!
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
Categories: Expressions ·

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