SYNOPSIS
package require Tcl 8.2package require math ?1.2.3?
::math::ln_Gamma z
::math::factorial x
::math::choose n k
::math::Beta z w
DESCRIPTION
The math package contains implementations of several functions useful in combinatorial problems.
COMMANDS
 ::math::ln_Gamma z

Returns the natural logarithm of the Gamma function for the argument
z.
The Gamma function is defined as the improper integral from zero to positive infinity of

t**(x1)*exp(t) dt
The approximation used in the Tcl Math Library is from Lanczos, ISIAM J. Numerical Analysis, series B, volume 1, p. 86. For "x > 1", the absolute error of the result is claimed to be smaller than 5.5*10**10  that is, the resulting value of Gamma when

exp( ln_Gamma( x) )

 is computed is expected to be precise to better than nine significant figures.
 ::math::factorial x

Returns the factorial of the argument x.
For integer x, 0 <= x <= 12, an exact integer result is returned.
For integer x, 13 <= x <= 21, an exact floatingpoint result is returned on machines with IEEE floating point.
For integer x, 22 <= x <= 170, the result is exact to 1 ULP.
For real x, x >= 0, the result is approximated by computing Gamma(x+1) using the ::math::ln_Gamma function, and the result is expected to be precise to better than nine significant figures.
It is an error to present x <= 1 or x > 170, or a value of x that is not numeric.
 ::math::choose n k

Returns the binomial coefficient C(n, k)

C(n,k) = n! / k! (nk)!


If both parameters are integers and the result fits in 32 bits, the
result is rounded to an integer.
Integer results are exact up to at least n = 34. Floating point results are precise to better than nine significant figures.
 ::math::Beta z w

Returns the Beta function of the parameters z and w.

Beta(z,w) = Beta(w,z) = Gamma(z) * Gamma(w) / Gamma(z+w)

 Results are returned as a floating point number precise to better than nine significant digits provided that w and z are both at least 1.