AngouriMath

Symbolics for .NET


Table of contents

AngouriMath

Navigation
Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug Donate Contacts

← Back to list of members

StirlingExponent​(AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​AngouriMath.​Entity)

 Method (no overloads)

Summary

power * ln(base) with the logarithm of every diverging factorial in it
replaced by Stirling's expansion, or null where there is no such
factorial or the expansion would not be sound here.

Remarks

ln(f!) is f*ln(f) - f + ln(2*pi*f)/2 + 1/(12f) + O(1/f^3), and what is
dropped **vanishes** -- where the asymptotic for f! itself has an error that
is merely relative. That is why the expansion is written for the logarithm and
applied to this exponent rather than substituted for the factorial in the base.
Vanishing is still not sufficient, because the dropped term is multiplied by the
exponent the rewrite sits under: the answer is e^(power * ln(base)), so an
error of 1/(12f) in the logarithm contributes power/(12f) to the
exponent. Requiring power / f -> 0 is what makes it disappear. For
((x!) / x^x)^(1/x) that ratio is 1/x^2.
The logarithm has to be taken apart before the factorial's own is visible:
ln(x!/x^x) is one node, and nothing here simplifies it. Splitting it over
products, quotients and powers assumes the parts are positive on the approach, which
is the same assumption the simplifier's ln(a) + ln(b) = ln(a*b) already
makes; it is confined to logarithms that actually hold a diverging factorial, so it
is reached only by expressions that have no answer at all without it.
#754

























Angouri © 2019-2023 · Project's repo · Site's repo · Octicons · Transparency · 1953 pages online