In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the...
12 KB (1,960 words) - 21:23, 15 May 2024
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that...
17 KB (2,773 words) - 08:12, 22 May 2024
Error function (section Asymptotic expansion)
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...
45 KB (6,818 words) - 08:15, 8 September 2024
In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation,...
20 KB (3,217 words) - 23:44, 19 August 2024
and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral. This implies e.g...
6 KB (1,104 words) - 21:51, 7 August 2024
Euler's constant (section Asymptotic expansions)
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...
52 KB (7,665 words) - 16:28, 8 September 2024
Polygamma function (section Asymptotic expansion)
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...
12 KB (2,364 words) - 05:12, 9 August 2024
Lerch zeta function (section Asymptotic expansion)
\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...
16 KB (3,490 words) - 09:53, 24 April 2024
Hermite polynomials (section Asymptotic expansion)
_{-\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...
57 KB (10,024 words) - 18:31, 4 August 2024
Digamma function (section Asymptotic expansion)
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 ∞ (...
35 KB (7,084 words) - 00:30, 21 August 2024
Big O notation (redirect from Asymptotic notation)
be used. Asymptotic computational complexity Asymptotic expansion: Approximation of functions generalizing Taylor's formula Asymptotically optimal algorithm:...
64 KB (8,280 words) - 04:39, 8 September 2024
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...
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Euler–Maclaurin formula (category Asymptotic analysis)
using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum...
19 KB (3,779 words) - 22:27, 25 March 2024
Stirling's approximation (category Asymptotic analysis)
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate...
27 KB (4,940 words) - 03:24, 30 August 2024
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...
11 KB (1,437 words) - 14:18, 7 August 2024
Stokes phenomenon (category Asymptotic analysis)
looking at the asymptotic expansion of an analytic function. Since an analytic function is continuous you would expect the asymptotic expansion to be continuous...
9 KB (1,323 words) - 10:28, 27 December 2023
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:...
26 KB (2,197 words) - 07:53, 18 July 2024
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...
92 KB (12,566 words) - 06:51, 6 September 2024
Taylor series (redirect from Taylor expansion)
distributed along the domain of the function. Mathematics portal Asymptotic expansion Newton polynomial Padé approximant – best approximation by a rational...
48 KB (8,253 words) - 02:02, 23 August 2024
The Cornish–Fisher expansion is an asymptotic expansion used to approximate the quantiles of a probability distribution based on its cumulants. It is named...
5 KB (738 words) - 16:43, 4 September 2024
direction gradually overtakes the Hubble flow of cosmic expansion in that direction, asymptotically approaching material with the same velocity as its own...
53 KB (6,980 words) - 16:28, 21 August 2024
Barnes G-function (section Asymptotic expansion)
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...
14 KB (2,634 words) - 19:01, 18 August 2024
Method of steepest descent (category Asymptotic analysis)
2307/2946540, JSTOR 2946540, S2CID 12699956. Erdelyi, A. (1956), Asymptotic Expansions, Dover. Fedoryuk, M. V. (2001) [1994], "Saddle point method", Encyclopedia...
31 KB (5,057 words) - 12:20, 17 April 2024
Multinomial theorem (redirect from Quadranomial expansion)
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...
9 KB (2,047 words) - 18:16, 24 August 2024
Bessel function (section Asymptotic forms)
sin ( x ) {\displaystyle \sin(x)} , as explicitly shown in the asymptotic expansion. The Hankel functions are used to express outward- and inward-propagating...
71 KB (11,575 words) - 07:49, 12 September 2024
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...
17 KB (2,900 words) - 18:53, 16 July 2024
expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which...
27 KB (4,364 words) - 22:48, 7 August 2024
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...
15 KB (2,030 words) - 19:21, 3 September 2024
Witten's asymptotic expansion conjecture suggests that for every 3-manifold M {\displaystyle M} , the large r {\displaystyle r} -th asymptotics of RT r...
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> 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)...
32 KB (7,714 words) - 15:50, 15 August 2024