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...

{\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...

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}}\left({\frac...

{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...

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...

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...

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...

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

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...

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...

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

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...

\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...

{\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...

}}\,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:...

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

{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 \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 ⁡...

hyperoctahedral groups (signed permutations or symmetries of a hypercube) Stirling's approximation for the factorial can be used to derive an asymptotic equivalent...

k}p^{k}(1-p)^{n-k}\simeq {\frac {\lambda ^{k}e^{-\lambda }}{k!}}.} Using Stirling's approximation, it can be written: ( n k ) p k ( 1 − p ) n − k = n ! ( n − k )...

{\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...

Θ ( n log ⁡ n ) {\displaystyle \log(n!)=\Theta (n\log n)} , by Stirling's approximation. They also frequently arise from the recurrence relation T ( n...

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

{\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...

{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...

− 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...

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...