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• Functions and Variables for stirling: |
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Replace gamma(x)
with the O(1/x^(2n-1)) Stirling formula. when n isn’t
a nonnegative integer, signal an error. With the optional third argument pred
,
the Stirling formula is applied only when pred
is true.
Reference: Abramowitz & Stegun, " Handbook of mathematical functions", 6.1.40.
Examples:
(%i1) load ("stirling")$ (%i2) stirling(gamma(%alpha+x)/gamma(x),1); 1/2 - x x + %alpha - 1/2 (%o2) x (x + %alpha) 1 1 --------------- - ---- - %alpha 12 (x + %alpha) 12 x %e (%i3) taylor(%,x,inf,1); %alpha 2 %alpha %alpha x %alpha - x %alpha (%o3)/T/ x + -------------------------------- + . . . 2 x (%i4) map('factor,%); %alpha - 1 %alpha (%alpha - 1) %alpha x (%o4) x + ------------------------------- 2
The function stirling
knows the difference between the variable ’gamma’ and
the function gamma:
(%i5) stirling(gamma + gamma(x),0); x - 1/2 - x (%o5) gamma + sqrt(2) sqrt(%pi) x %e (%i6) stirling(gamma(y) + gamma(x),0); y - 1/2 - y (%o6) sqrt(2) sqrt(%pi) y %e x - 1/2 - x + sqrt(2) sqrt(%pi) x %e
To apply the Stirling formula only to terms that involve the variable k
,
use an optional third argument; for example
(%i7) makegamma(pochhammer(a,k)/pochhammer(b,k)); (%o7) (gamma(b)*gamma(k+a))/(gamma(a)*gamma(k+b)) (%i8) stirling(%,1, lambda([s], not(freeof(k,s)))); (%o8) (%e^(b-a)*gamma(b)*(k+a)^(k+a-1/2)*(k+b)^(-k-b+1/2))/gamma(a)
The terms gamma(a)
and gamma(b)
are free of k
, so the Stirling formula
was not applied to these two terms.
To use this function write first load("stirling")
.
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