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The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.
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.
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.
Complex bigfloat factorial.
load ("bffac") loads this function.
The basic definition of the gamma function (A&S eqn 6.1.1) is
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.
The natural logarithm of the gamma function.
The lower incomplete gamma function (A&S eqn 6.5.2):
See also gamma_incomplete (upper incomplete gamma function).
The incomplete upper gamma function (A&S eqn 6.5.3):
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.
The regularized incomplete upper gamma function (A&S eqn 6.5.1):
See also gamma_expand for controlling how
gamma_incomplete is expressed in terms of elementary functions
and erfc.
Also see gamma_incomplete.
The generalized incomplete gamma function.
Also see gamma_incomplete and gamma_incomplete_regularized.
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(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
(%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
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.
Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.
See also makefact.
The beta function is defined as
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);
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
The basic definition of the incomplete beta function (A&S eqn 6.6.1) is
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
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)
The regularized incomplete beta function (A&S eqn 6.6.2), defined as
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
The basic definition of the generalized incomplete beta function is
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)
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.
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.
psi[n](x) is defined by
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.
Default value: 20
maxpsiposint is the largest positive value for which
psi[n](x) will try to compute an exact value.
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.
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.
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.
Transforms instances of binomial, gamma, and beta functions in expr into factorials.
See also makegamma.
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
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