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searching for Asymptotic expansion 76 found (134 total)

Stokes phenomenon (1,322 words) [view diff] exact match in snippet view article find links to article

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

Derangement (2,203 words) [view diff] exact match in snippet view article find links to article

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:

Parabolic cylinder function (2,655 words) [view diff] exact match in snippet view article find links to article

{1}{8}}t^{4}-\dots \;} in the integrand of the integral representation gives the asymptotic expansion of U ( a , z ) {\displaystyle U(a,z)} , U ( a , z ) = e − 1 4 z 2

Topological derivative (1,725 words) [view diff] exact match in snippet view article find links to article

gradient is also used to name the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations. It

Barnes G-function (2,937 words) [view diff] exact match in snippet view article find links to article

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

Digamma function (7,244 words) [view diff] exact match in snippet view article find links to article

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 ∞ (

Reshetikhin–Turaev invariant (1,656 words) [view diff] exact match in snippet view article find links to article

Pick t = e π i r {\displaystyle t=e^{\frac {\pi i}{r}}} . Witten's asymptotic expansion conjecture suggests that for every 3-manifold M {\displaystyle M}

Lerch transcendent (3,654 words) [view diff] exact match in snippet view article find links to article

\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

Riemann–Siegel formula (857 words) [view diff] exact match in snippet view article find links to article

applying the method of steepest descent to this integral to give an asymptotic expansion for the error term R(s) as a series of negative powers of Im(s).

Polygamma function (2,386 words) [view diff] no match in snippet view article find links to article

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

Sierpiński's constant (406 words) [view diff] exact match in snippet view article find links to article

Summatory Function of r 2 ( k ) / k {\displaystyle r_{2}(k)/k} has the Asymptotic expansion ∑ k = 1 n r 2 ( k ) k = K + π ln ⁡ n + o ( 1 n ) {\displaystyle \sum

Reciprocal gamma function (1,431 words) [view diff] no match in snippet view article find links to article

In mathematics, the reciprocal gamma function is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the

Minakshisundaram–Pleijel zeta function (1,145 words) [view diff] exact match in snippet view article find links to article

Asymptotic Expansion Let (M,g) be an n-dimensional Riemannian manifold. Then, as t→0+, the trace of the heat kernel has an asymptotic expansion of

Padé approximant (2,999 words) [view diff] exact match in snippet view article find links to article

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

Steven Zelditch (1,039 words) [view diff] exact match in snippet view article find links to article

manifold. The Tian-Yau-Zelditch theorem in this case gives a complete asymptotic expansion of the Bergman kernel near the diagonal. For example, the Catlin-De

Method of steepest descent (5,062 words) [view diff] no match in snippet view article find links to article

In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms

Erlang distribution (1,969 words) [view diff] exact match in snippet view article find links to article

the same as f ( x ; 2 , λ ) {\displaystyle f(x;2,\lambda )} . An asymptotic expansion is known for the median of an Erlang distribution, for which coefficients

K-nearest neighbors algorithm (4,333 words) [view diff] exact match in snippet view article find links to article

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 ) {

Trigamma function (1,125 words) [view diff] exact match in snippet view article find links to article

{x^{z-1}\ln {x}}{1-x}}\,dx} An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:

Bessel–Maitland function (172 words) [view diff] exact match in snippet view article find links to article

{(-z)^{k}}{\Gamma (k\mu +\nu +1)k!}}.} Wright, E. M. (1934), "The asymptotic expansion of the generalized Bessel function.", Proceedings of the London Mathematical

Meixner polynomials (518 words) [view diff] exact match in snippet view article find links to article

doi:10.1006/jmaa.1994.1214. Jin, X.-S.; Wong, R. (1998). "Uniform asymptotic expansion for Meixner polynomials". Construct. Approx. 14 (1): 113–150. doi:10

Uniform theory of diffraction (218 words) [view diff] case mismatch in snippet view article find links to article

and Propagation Society International Symposium. doi:10.1109/APS.2003.1220345. Overview of Asymptotic Expansion Methods in Electromagnetics v t e v t e

Weyl law (859 words) [view diff] exact match in snippet view article find links to article

Inventiones Mathematicae, 29(1):37–79 (1975). Second term of the spectral asymptotic expansion for the Laplace–Beltrami operator on manifold with boundary. Functional

Konrad Jörgens (251 words) [view diff] exact match in snippet view article find links to article

of applied mathematics at Heidelberg. Jörgens, Konrad (1958). "An asymptotic expansion in the theory of neutron transport". Comm. Pure Appl. Math. 11 (2):

J-integral (1,996 words) [view diff] exact match in snippet view article find links to article

with origin at the crack tip, s is a constant determined from an asymptotic expansion of the stress field around the crack, and I is a dimensionless integral

Topological recursion (4,390 words) [view diff] exact match in snippet view article find links to article

find the large size asymptotic expansion of n-point correlation functions, and in some suitable cases, the asymptotic expansion takes the form of a power

Yukawa potential (4,261 words) [view diff] exact match in snippet view article find links to article

ν {\displaystyle \nu } that replaces the integer n is really an asymptotic expansion like that above with first approximation the integer value of the

Boris Rufimovich Vainberg (237 words) [view diff] case mismatch in snippet view article find links to article

University Press, 2002 (with N. Kuznetsov and V. Maz'ya) Large Time Asymptotic Expansion of the Solutions of Exterior Boundary Value Problems for Hyperbolic

Alice Guionnet (1,070 words) [view diff] exact match in snippet view article find links to article

aim.2009.03.019. S2CID 18359348. G. Borot & A. Guionnet (2013). "Asymptotic expansion of beta matrix models in the one-cut regime". Communications in Mathematical

Wedderburn–Etherington number (1,473 words) [view diff] exact match in snippet view article find links to article

1016/0166-218X(86)90068-5, MR 0829338. Landau, B. V. (1977), "An asymptotic expansion for the Wedderburn-Etherington sequence", Mathematika, 24 (2): 262–265

Gregory coefficients (2,506 words) [view diff] exact match in snippet view article find links to article

numbers. Discrete Math., vol. 306, pp. 1906–1920, 2006. G. Nemes. An asymptotic expansion for the Bernoulli numbers of the second kind. J. Integer Seq, vol

Perturbation theory (2,959 words) [view diff] exact match in snippet view article find links to article

perturbation series is divergent or not a power series (for example, if the asymptotic expansion must include non-integer powers ε ( 1 / 2 ) {\displaystyle \

Dickman function (1,206 words) [view diff] exact match in snippet view article find links to article

constants". arXiv:1004.0519 [math-ph]. Soundararajan, Kannan (2012). "An asymptotic expansion related to the Dickman function". Ramanujan Journal. 29 (1–3): 25–30

Totient summatory function (647 words) [view diff] exact match in snippet view article find links to article

n ) {\displaystyle \mu (n)} is the Möbius function. Φ(n) has the asymptotic expansion Φ ( n ) ∼ 1 2 ζ ( 2 ) n 2 + O ( n log ⁡ n ) = 3 π 2 n 2 + O ( n log

Rice distribution (3,232 words) [view diff] exact match in snippet view article find links to article

the large x ν / σ 2 {\displaystyle x\nu /\sigma ^{2}} region, an asymptotic expansion of the Rician distribution: f ( x , ν , σ ) = x σ 2 exp ⁡ ( − ( x

Wilks's lambda distribution (635 words) [view diff] case mismatch in snippet view article find links to article

Soc Ser B. 16 (2): 296–298. JSTOR 2984057. C. R. Rao (1951). "An Asymptotic Expansion of the Distribution of Wilks' Criterion". Bulletin de l'Institut

Dawson function (1,626 words) [view diff] exact match in snippet view article find links to article

{4}{15}}x^{5}-\cdots ,} while for large x {\displaystyle x} it has the asymptotic expansion F ( x ) = 1 2 x + 1 4 x 3 + 3 8 x 5 + ⋯ . {\displaystyle F(x)={\frac

Stieltjes transformation (1,170 words) [view diff] exact match in snippet view article find links to article

the Stieltjes transformation of ρ admits for each integer n the asymptotic expansion in the neighbourhood of infinity given by S ρ ( z ) = ∑ k = 0 n m

Hahn–Exton q-Bessel function (893 words) [view diff] exact match in snippet view article find links to article

converges fast at x → ∞ {\displaystyle x\to \infty } . It is also an asymptotic expansion for ν → ∞ {\displaystyle \nu \to \infty } . Abreu, L. D.; Bustoz

Witten conjecture (1,167 words) [view diff] exact match in snippet view article find links to article

By Feynman diagram techniques, this implies that F(t0,...) is an asymptotic expansion of log ⁡ ∫ exp ⁡ ( i tr X 3 / 6 ) d μ {\displaystyle \log \int

Holonomic function (1,977 words) [view diff] exact match in snippet view article find links to article

and special functions (evaluation at a point, Taylor series and asymptotic expansion to any user-given precision, differential equation, recurrence for

Entire function (3,297 words) [view diff] exact match in snippet view article find links to article

any | z | ≥ 1 . {\displaystyle |z|\geq 1~.} Boas 1954, p. 1. See asymptotic expansion in Abramowitz and Stegun, p. 377, 9.7.1. Boas, Ralph P. (1954). Entire

Lamé function (1,793 words) [view diff] exact match in snippet view article find links to article

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

Holstein–Herring method (2,003 words) [view diff] exact match in snippet view article find links to article

Conyers Herring was the first to show that the lead term for the asymptotic expansion of the energy difference between the two lowest states of the hydrogen

Hakan Hedenmalm (640 words) [view diff] exact match in snippet view article find links to article

functions. Advances in Mathematics 264 (2014), 464-505. (with A. Haimi) Asymptotic expansion of polyanalytic Bergman kernels. Journal of Functional Analysis 267

Xiaonan Ma (906 words) [view diff] exact match in snippet view article find links to article

doi:10.1090/S0002-9947-05-03847-X. Dai, X.; Liu, K.; Ma, X. (2006). "Asymptotic expansion of the Bergman kernel". Journal of Differential Geometry. 72: 1–41

Happy ending problem (1,877 words) [view diff] exact match in snippet view article find links to article

used here and Catalan numbers or Stirling's approximation for the asymptotic expansion. Holmsen et al. (2020). Harborth (1978). Horton (1983) Overmars (2003)

Bose gas (3,023 words) [view diff] exact match in snippet view article find links to article

the method of inversion of series, either in powers of τα or as an asymptotic expansion in inverse powers of τα. From these expansions, we can find the behavior

Multiple-scale analysis (1,870 words) [view diff] exact match in snippet view article find links to article

Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution. To construct a solution that is valid beyond t =

1 + 2 + 3 + 4 + ⋯ (4,170 words) [view diff] exact match in snippet view article find links to article

where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f: it is necessarily the same value given by analytic

Inertial manifold (1,941 words) [view diff] exact match in snippet view article find links to article

This differential equation is typically solved approximately in an asymptotic expansion in 'small' p {\displaystyle p} to give an invariant manifold model

Tracy–Widom distribution (5,053 words) [view diff] exact match in snippet view article find links to article

{3}{2^{6}x^{3}}}+O(x^{-6}){\biggr )}.\end{aligned}}} The Painlevé transcendent has asymptotic expansion at x → − ∞ {\displaystyle x\to -\infty } (equation 4.1 of ) q ( x

Stirling numbers of the second kind (4,328 words) [view diff] case mismatch in snippet view article find links to article

1016/S0021-9800(69)80045-1. ISSN 0021-9800. L. C. Hsu, Note on an Asymptotic Expansion of the nth Difference of Zero, AMS Vol.19 NO.2 1948, pp. 273--277

J. Ernest Wilkins Jr. (2,295 words) [view diff] case mismatch in snippet view article find links to article

Functions", Proceedings of the American Mathematical Society, 1985; "An Asymptotic Expansion for the Expected Number of Real Zeros of a Random Polynomial", Proceedings

Harmonic series (mathematics) (6,357 words) [view diff] exact match in snippet view article

convergent. Its sum is the natural logarithm of 2. More precisely, the asymptotic expansion of the series begins as 1 1 − 1 2 + ⋯ + 1 2 n − 1 − 1 2 n = H 2 n

Tian Gang (3,406 words) [view diff] exact match in snippet view article find links to article

Boston, Boston, MA, 1999. Lu, Zhiqin. On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math. 122 (2000), no. 2, 235–273.

Yang–Baxter equation (3,429 words) [view diff] exact match in snippet view article find links to article

quantum YBE, in which the R {\displaystyle R} -matrix admits an asymptotic expansion in terms of an expansion parameter ℏ , {\displaystyle \hbar ,} R

Yang–Baxter equation (3,429 words) [view diff] exact match in snippet view article find links to article

quantum YBE, in which the R {\displaystyle R} -matrix admits an asymptotic expansion in terms of an expansion parameter ℏ , {\displaystyle \hbar ,} R

Faxén integral (592 words) [view diff] exact match in snippet view article find links to article

Flüssigkeit (PhD). Uppsala University. Paris, Richard Bruce (2010). "Asymptotic expansion of n-dimensional Faxén-type integrals". European Journal of Pure

Fresnel integral (2,945 words) [view diff] exact match in snippet view article find links to article

{3}{2}}\end{array}}\mid -{\frac {x^{2n}}{4}}\right).} The leading term in the asymptotic expansion is 1 F 1 ( m + 1 n 1 + m + 1 n ∣ i x n ) ∼ m + 1 n Γ ( m + 1 n )

Gamma distribution (8,873 words) [view diff] exact match in snippet view article find links to article

{1}{2}}.} A rigorous treatment of the problem of determining an asymptotic expansion and bounds for the median of the gamma distribution was handled first

Atiyah–Singer index theorem (7,553 words) [view diff] exact match in snippet view article find links to article

the difference of the kernels of two heat operators. These have an asymptotic expansion for small positive t, which can be used to evaluate the limit as

Central limit theorem (9,208 words) [view diff] exact match in snippet view article find links to article

popular tools employed to approach such questions. Suppose we have an asymptotic expansion of f ( n ) {\textstyle f(n)} : f ( n ) = a 1 φ 1 ( n ) + a 2 φ 2

Euler's constant (9,613 words) [view diff] exact match in snippet view article find links to article

product formula for the gamma function and the Barnes G-function. The asymptotic expansion of the gamma function, Γ ( 1 / x ) ∼ x − γ {\displaystyle \Gamma

Percolation critical exponents (6,649 words) [view diff] exact match in snippet view article find links to article

exponents extend these sets, they refer to higher orders of the asymptotic expansion around the critical point. Percolation clusters become self-similar

Jackson q-Bessel function (2,730 words) [view diff] exact match in snippet view article find links to article

&0\\-{\sqrt {q}},&iax\end{matrix}};{\sqrt {q}},{\sqrt {q}}\right).} An asymptotic expansion can be obtained as an immediate consequence of the second formula

Michael Atiyah (8,796 words) [view diff] exact match in snippet view article find links to article

the stationary phase approximation (which is in general just an asymptotic expansion rather than exact). Atiyah and Bott showed that this could be deduced

Modified half-normal distribution (1,797 words) [view diff] case mismatch in snippet view article find links to article

ISSN 2524-0021. S2CID 242771336. Wright, E. Maitland (1935). "The Asymptotic Expansion of the Generalized Hypergeometric Function". Journal of the London

Hankel transform (4,210 words) [view diff] exact match in snippet view article find links to article

methods based on the projection-slice theorem, and methods using the asymptotic expansion of Bessel functions. Kn(z) is a modified Bessel function of the second

Incomplete gamma function (7,107 words) [view diff] exact match in snippet view article find links to article

ISBN 978-0-521-19225-5, MR 2723248. Paris, R. B. (2002). "A uniform asymptotic expansion for the incomplete gamma function". J. Comput. Appl. Math. 148 (2):

Dihydrogen cation (3,809 words) [view diff] exact match in snippet view article find links to article

eigenenergies Eg/u for these two lowest lying states have the same asymptotic expansion in inverse powers of the internuclear distance R: E g / u = − 1 2

Radial distribution function (4,537 words) [view diff] exact match in snippet view article find links to article

S2CID 254274307. Tricomi, F.; Erdélyi, A. (March 1, 1951). "The asymptotic expansion of a ratio of gamma functions". Pacific Journal of Mathematics. 1

Hartman–Watson distribution (1,355 words) [view diff] exact match in snippet view article find links to article

efficient method for the numerical evaluation of this integral uses an asymptotic expansion as t→0 in the limit r t = ρ constant obtained from a saddle point

Frederic Wan (2,499 words) [view diff] exact match in snippet view article find links to article

latter category would be the determination of the interior (or outer-asymptotic expansion) solution for a BVP (for a plate or shell structure) independent

Random permutation statistics (11,999 words) [view diff] exact match in snippet view article find links to article

using an integral estimate such as Euler–Maclaurin summation, or the asymptotic expansion of the nth harmonic number, we obtain H 2 n − H n ∼ log ⁡ 2 − 1 4

Plancherel theorem for spherical functions (11,769 words) [view diff] exact match in snippet view article find links to article

expressed in terms of the hypergeometric function for which the asymptotic expansion is known from the classical formulas of Gauss for the connection