In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the...

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that...

In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation,...

this asymptotic expansion are needed to obtain a good approximation of erfc x (while for not too large values of x, the above Taylor expansion at 0 provides...

and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral. This implies e.g...

the digamma function A product formula for the gamma function The asymptotic expansion of the gamma function for small arguments. An inequality for Euler's...

In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m + 1)th...

integral representation can be manipulated to give the start of the asymptotic expansion of ψ {\displaystyle \psi } . ψ ( z ) = log ⁡ z − 1 2 z − ∫ 0 ∞ (...

\mathbb {C} } and z ∈ Ω a {\displaystyle z\in \Omega _{a}} , an asymptotic expansion of Φ ( z , s , a ) {\displaystyle \Phi (z,s,a)} for large a {\displaystyle...

Euler and Weierstrass respectively, we get the following infinite product expansion for the reciprocal gamma function: 1 Γ ( z ) = z ∏ n = 1 ∞ 1 + z n ( 1...

_{-\infty }^{\infty }(x+iy)^{n}e^{-{\frac {y^{2}}{2}}}\,dy.} Asymptotically, as n → ∞, the expansion e − x 2 2 ⋅ H n ( x ) ∼ 2 n π Γ ( n + 1 2 ) cos ⁡ ( x 2...

using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum...

described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for...

be used. Asymptotic computational complexity Asymptotic expansion: Approximation of functions generalizing Taylor's formula Asymptotically optimal algorithm:...

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

found in the article on the statistics of random permutations. An asymptotic expansion for the number of derangements in terms of Bell numbers is as follows:...

looking at the asymptotic expansion of an analytic function. Since an analytic function is continuous you would expect the asymptotic expansion to be continuous...

> 0; and that the expansion of f can be term wise differentiated. Then, Laplace–Erdelyi theorem states that the asymptotic expansion of the integral I(λ)...

identity reduces to 1 = 1 , {\displaystyle 1=1,} so we must look at its asymptotic expansion. The Maclaurin series for the cosine function can be written as cos...

series the error is distributed along the domain of the function. Asymptotic expansion Generating function Laurent series Madhava series Newton's divided...

2307/2946540, JSTOR 2946540, S2CID 12699956. Erdelyi, A. (1956), Asymptotic Expansions, Dover. Fedoryuk, M. V. (2001) [1994], "Saddle point method", Encyclopedia...

Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion, log ⁡ ( k n n , n , ⋯ , n ) = k n log ⁡ ( k ) + 1 2 ( log ⁡ ( k...

multi-point summation method. Since there are many cases in which the asymptotic expansion at infinity becomes 0 or a constant, it can be interpreted as the...

The Cornish–Fisher expansion is an asymptotic expansion used to approximate the quantiles of a probability distribution based on its cumulants. It is named...

criteria. On the class distributions the excess risk has the following asymptotic expansion R R ( C n w n n ) − R R ( C Bayes ) = ( B 1 s n 2 + B 2 t n 2 ) {...

Witten's asymptotic expansion conjecture suggests that for every 3-manifold M {\displaystyle M} , the large r {\displaystyle r} -th asymptotics of RT r...

values of κ {\displaystyle \kappa } have been obtained by Müller. The asymptotic expansion obtained by him for the eigenvalues Λ {\displaystyle \Lambda } is...

low strike, short expiry, large expiry) it is possible to give an asymptotic expansion of implied volatility in terms of call price. A different approach...

sin ⁡ ( x ) {\displaystyle \sin(x)} , as explicitly shown in the asymptotic expansion. The Hankel functions are used to express outward- and inward-propagating...

expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which...