mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate...
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Factorial (redirect from Approximations of factorial)
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...
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Gamma function (redirect from Approximations of the gamma function)
accurate approximation can be obtained by using more terms from the asymptotic expansions of ln(Γ(z)) and Γ(z), which are based on Stirling's approximation. Γ...
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Abraham de Moivre (section Stirling's approximation)
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...
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to the more popular Stirling's approximation for calculating the gamma function with fixed precision. The Lanczos approximation consists of the formula...
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{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...
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{\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...
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and complex integration. Laplace's method can be used to derive Stirling's approximation N ! ≈ 2 π N N N e − N {\displaystyle N!\approx {\sqrt {2\pi N}}N^{N}e^{-N}\...
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mathematician. He was nicknamed "The Venetian". The Stirling numbers, Stirling permutations, and Stirling's approximation are named after him. He also proved the...
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Asymptotic analysis (redirect from Asymptotic approximation)
\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...
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constant) Exponential function Hyperbolic angle Hyperbolic function Stirling's approximation Bernoulli numbers See also list of numerical analysis topics Rectangle...
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2k selected elements. It follows from the summation formula and Stirling's approximation that, asymptotically, T ( n ) ∼ ( n e ) n / 2 e n ( 4 e ) 1 / 4...
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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...
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postulate Sierpinski triangle Star of David theorem Stirling number Stirling transform Stirling's approximation Subfactorial Table of Newtonian series Taylor...
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Look up Stirling in Wiktionary, the free dictionary. Stirling is a city and former ancient burgh in Scotland. Stirling may also refer to: Stirling's approximation...
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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...
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Chi distribution (section Large n approximation)
}}\,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:...
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nO(\log n)-O(n)=O(n\log n)} . This can also be readily seen from Stirling's approximation. Amortized time. Brodal and Okasaki describe a technique to reduce...
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may be computed using a generalization of Kummer's theorem. By Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion...
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Beta function (section Approximation)
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...
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{\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}...
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{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}...
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hyperoctahedral groups (signed permutations or symmetries of a hypercube) Stirling's approximation for the factorial can be used to derive an asymptotic equivalent...
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{\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...
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Park. Mathematics portal Binomial approximation Binomial distribution Binomial inverse theorem Stirling's approximation Tannery's theorem Polynomials calculating...
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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...
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{\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...
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Inverse gamma function (section Approximation)
is the Lambert W function. The formula is found by inverting the Stirling approximation, and so can also be expanded into an asymptotic series. To obtain...
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numerator of the fraction would grow singly exponentially while by Stirling's approximation the denominator grows more quickly than singly exponentially),...
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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 )...
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