mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate...

an approximation for the central term of a binomial expansion. (de Moivre, 1730), p. 99. The roles of de Moivre and Stirling in finding Stirling's approximation...

to the more popular Stirling's approximation for calculating the gamma function with fixed precision. The Lanczos approximation consists of the formula...

{4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdots \end{aligned}}}     Stirling's approximation for the factorial function n ! {\displaystyle n!} asserts that...

{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} which is known as Stirling's approximation. Equivalently, π = lim n → ∞ e 2 n n ! 2 2 n 2 n + 1 . {\displaystyle...

accurate approximation can be obtained by using more terms from the asymptotic expansions of logΓ(z) and Γ(z), which are based on Stirling's approximation. Γ...

and complex integration. Laplace's method can be used to derive Stirling's approximation N ! ≈ 2 π N ( N e ) N {\displaystyle N!\approx {\sqrt {2\pi N}}({\frac...

Moivre in 1721, a 1729 letter from James Stirling to de Moivre stating what became known as Stirling's approximation, and work at the same time by Daniel...

given via Stirling's approximation. An upper bound of the same form, with the same leading term as the bound obtained from Stirling's approximation, follows...

mathematician. He was nicknamed "The Venetian". The Stirling numbers, Stirling permutations, and Stirling's approximation are named after him. He also proved the...

the formula in a 1994 paper. The formula is a modification of Stirling's approximation, and has the form Γ ( z + 1 ) = ( z + a ) z + 1 2 e − z − a ( c...

constant) Exponential function Hyperbolic angle Hyperbolic function Stirling's approximation Bernoulli numbers See also list of numerical analysis topics Rectangle...

\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} —this is Stirling's approximation Partition function For a positive integer n, the partition function...

nO(\log n)-O(n)=O(n\log n)} . This can also be readily seen from Stirling's approximation. make-heap is the operation of building a heap from a sequence...

L   {\displaystyle \ L\ } possibly infinite). Gamma function (Stirling's approximation) e x x x 2 π x Γ ( x + 1 ) ∼ 1 + 1 12 x + 1 288 x 2 − 139 51840...

postulate Sierpinski triangle Star of David theorem Stirling number Stirling transform Stirling's approximation Subfactorial Table of Newtonian series Taylor...

{\displaystyle \exp({P_{k}(n)})} . For k = 0 {\displaystyle k=0} we get Stirling's approximation without the factor 2 π {\displaystyle {\sqrt {2\pi }}} as exp ⁡...

{\displaystyle \mu _{g}=-k_{\rm {B}}T\ln(q/N)} , where we use Stirling's approximation. Plugging μ g {\displaystyle \mu _{g}} to the expression of x {\displaystyle...

Mathematics portal Binomial approximation Binomial distribution Binomial inverse theorem Binomial coefficient Stirling's approximation Tannery's theorem Polynomials...

explained in the article Comparison sort) and in case of large n, Stirling's approximation yields log2(n!) ≈ n(log2 n − log2 e), so quicksort is not much...

{\displaystyle g_{i}\gg N_{i}} . Under these conditions, we may use Stirling's approximation for the factorial: N ! ≈ N N e − N , {\displaystyle N!\approx N^{N}e^{-N}...

{3}{2}}\right]+{\frac {5}{2}}\end{aligned}}} The derivation uses Stirling's approximation, ln ⁡ N ! ≈ N ln ⁡ N − N {\displaystyle \ln N!\approx N\ln N-N}...

{\displaystyle n\to \infty } . In this case, one may make use of Stirling's approximation to the factorial, and write n ! = 2 π n n n e − n ( 1 + O ( 1 n...

{R(k)}{T_{u}^{k}}}-\Theta _{v}\,{\frac {R(v+1)}{T_{u}^{v+1}}}\end{aligned}}} From Stirling's approximation follows a similar series: γ = ln ⁡ 2 π − ∑ k = 2 ∞ ζ ( k ) T k...

{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} (Stirling's approximation) log ⁡ n ! ≃ ( n + 1 2 ) log ⁡ n − n + log ⁡ 2 π 2 {\displaystyle...

}}\,2^{n-2}\,{\frac {(\Gamma (n/2))^{2}}{\Gamma (n-1)}}} Using Stirling's approximation for Gamma function, we get the following expression for the mean:...

− 1 {\displaystyle 1\leq k\leq n-1} .: 309  Stirling's approximation yields the following approximation, valid when n − k , k {\displaystyle n-k,k} both...

initial state is finite, and it is 2 N {\displaystyle 2^{N}} . Using Stirling's approximation one finds that if we start at equilibrium (equal number of particles...

ψ ( z ) {\displaystyle \psi (z)} denotes the digamma function. Stirling's approximation gives the asymptotic formula B ( x , y ) ∼ 2 π x x − 1 / 2 y y...

complex numbers Gamma function: Lanczos approximation Spouge's approximation — modification of Stirling's approximation; easier to apply than Lanczos AGM method...