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...

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...

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

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...

\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...

_{-\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...

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 ∞ (...

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

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...

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

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

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...

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

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:...

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...

distributed along the domain of the function. Mathematics portal Asymptotic expansion Newton polynomial Padé approximant – best approximation by a rational...

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

direction gradually overtakes the Hubble flow of cosmic expansion in that direction, asymptotically approaching material with the same velocity as its own...

of order two, and of infinite type. This can be deduced from the asymptotic expansion given below. The Barnes G-function satisfies the functional equation...

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...

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

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...

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

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...

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

> 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)...