Elementary Functions
Function: !!
The double factorial operator.
For an integer, float, or rational number n, n!! evaluates to the
product n (n-2) (n-4) (n-6) ... (n - 2 (k-1)) where k is equal to
entier (n/2), that is, the largest integer less than or equal to
n/2. Note that this definition does not coincide with other published
definitions for arguments which are not integers.
For an even (or odd) integer n, n!! evaluates to the product of
all the consecutive even (or odd) integers from 2 (or 1) through n
inclusive.
For an argument n which is not an integer, float, or rational, n!!
yields a noun form genfact (n, n/2, 2).
binomial (x, y) — Function
The binomial coefficient x!/(y! (x - y)!).
If x and y are integers, then the numerical value of the binomial
coefficient is computed. If y, or x - y, is an integer, the
binomial coefficient is expressed as a polynomial.
Examples:
maxima
(%i1) binomial (11, 7);
(%o1) 330
(%i2) 11! / 7! / (11 - 7)!;
(%o2) 330
(%i3) binomial (x, 7);
(x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3) -------------------------------------------------
5040
(%i4) binomial (x + 7, x);
(x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
5040
(%i5) binomial (11, y);
(%o5) binomial(11, y)
factcomb (expr) — Function
Tries to combine the coefficients of factorials in expr
with the factorials themselves by converting, for example, (n + 1)*n!
into (n + 1)!.
sumsplitfact if set to false will cause minfactorial to be
applied after a factcomb.
Example:
maxima
(%i1) sumsplitfact;
(%o1) true
(%i2) (n + 1)*(n + 1)*n!;
2
(%o2) (n + 1) n!
(%i3) factcomb (%);
(%o3) (n + 2)! - (n + 1)!
(%i4) sumsplitfact: not sumsplitfact;
(%o4) false
(%i5) (n + 1)*(n + 1)*n!;
2
(%o5) (n + 1) n!
(%i6) factcomb (%);
(%o6) n (n + 1)! + (n + 1)!
See also: sumsplitfact, minfactorial.
factlim — Variable
Default value: 100000
factlim specifies the highest factorial which is
automatically expanded. If it is -1 then all integers are expanded.
Function: factorial
Represents the factorial function. Maxima treats factorial (x)
the same as x!.
For any complex number x, except for negative integers, x! is
defined as gamma(x+1).
For an integer x, x! simplifies to the product of the integers
from 1 to x inclusive. 0! simplifies to 1. For a real or complex
number in float or bigfloat precision x, x! simplifies to the
value of gamma (x+1). For x equal to n/2 where n is
an odd integer, x! simplifies to a rational factor times
sqrt (%pi) (since gamma (1/2) is equal to sqrt (%pi)).
The option variables factlim and gammalim control the numerical
evaluation of factorials for integer and rational arguments. The functions
minfactorial and factcomb simplifies expressions containing
factorials.
The functions gamma, bffac, and cbffac are
varieties of the gamma function. bffac and cbffac are called
internally by gamma to evaluate the gamma function for real and complex
numbers in bigfloat precision.
makegamma substitutes gamma for factorials and related functions.
Maxima knows the derivative of the factorial function and the limits for specific values like negative integers.
The option variable factorial_expand controls the simplification of
expressions like (n+x)!, where n is an integer.
See also binomial.
The factorial of an integer is simplified to an exact number unless the operand
is greater than factlim. The factorial for real and complex numbers is
evaluated in float or bigfloat precision.
maxima
(%i1) factlim : 10;
(%o1) 10
(%i2) [0!, (7/2)!, 8!, 20!];
105 sqrt(%pi)
(%o2) [1, -------------, 40320, 20!]
16
(%i3) [4,77!, (1.0+%i)!];
(%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]
(%i4) [2.86b0!, (1.0b0+%i)!];
(%o4) [5.046635586910012b0, 3.430658398165454b-1 %i
+ 6.529654964201667b-1]
The factorial of a known constant, or general expression is not simplified. Even so it may be possible to simplify the factorial after evaluating the operand.
maxima
(%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
(%o1) [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
(%i2) ev (%, numer, %enumer);
(%o2) [0.3430658398165453 %i + 0.6529654964201667,
7.188082728976031, 4.260820476357003, 1.227580202486819]
Factorials are simplified, not evaluated.
Thus x! may be replaced even in a quoted expression.
maxima
(%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
105 sqrt(%pi)
(%o1) [1, -------------, 81.44668037931197, 40320,
16
2432902008176640000]
Maxima knows the derivative of the factorial function.
maxima
(%i1) diff(x!,x);
(%o1) x! psi (x + 1)
0
The option variable factorial_expand controls expansion and
simplification of expressions with the factorial function.
maxima
(%i1) (n+1)!/n!,factorial_expand:true;
(%o1) n + 1
See also: factlim, gammalim, minfactorial, factcomb, gamma, bffac, cbffac, makegamma, factorial_expand, binomial.
factorial_expand — Variable
Default value: false
The option variable factorial_expand controls the simplification of
expressions like (x+n)!, where n is an integer.
See factorial for an example.
See also: factorial.
genfact (x, y, z) — Function
Returns the generalized factorial, defined as
x (x-z) (x - 2 z) ... (x - (y - 1) z). Thus, when x is an integer,
genfact (x, x, 1) = x! and genfact (x, x/2, 2) = x!!.
When x and z are integers,
and floor(y) is an integer,
genfact(x, y, y) simplifies to a number.
minfactorial (expr) — Function
Examines expr for occurrences of two factorials
which differ by an integer.
minfactorial then turns one into a polynomial times the other.
maxima
(%i1) n!/(n+2)!;
n!
(%o1) --------
(n + 2)!
(%i2) minfactorial (%);
1
(%o2) ---------------
(n + 1) (n + 2)
sumsplitfact — Variable
Default value: true
When sumsplitfact is false,
minfactorial is applied after a factcomb.
maxima
(%i1) sumsplitfact;
(%o1) true
(%i2) n!/(n+2)!;
n!
(%o2) --------
(n + 2)!
(%i3) factcomb(%);
n!
(%o3) --------
(n + 2)!
(%i4) sumsplitfact: not sumsplitfact ;
(%o4) false
(%i5) n!/(n+2)!;
n!
(%o5) --------
(n + 2)!
(%i6) factcomb(%);
1
(%o6) ---------------
(n + 1) (n + 2)
See also: minfactorial, factcomb.