In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.
Theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886.[1]
The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.
Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics.[2] The error is then typically of the form ~ exp(−c/ε) where ε is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.
See asymptotic analysis and big O notation for the notation used in this article.
^Jahnke, Hans Niels (2003). A history of analysis. History of mathematics. Providence (R.I.): American mathematical society. p. 190. ISBN 978-0-8218-2623-2.
^Boyd, John P. (1999), "The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series" (PDF), Acta Applicandae Mathematicae, 56 (1): 1–98, doi:10.1023/A:1006145903624, hdl:2027.42/41670.
and 21 Related for: Asymptotic expansion information
In mathematics, an asymptoticexpansion, 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 asymptoticexpansions is a common approach to finding an accurate approximation to the solution to an equation,...
this asymptoticexpansion 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 asymptoticexpansion for the exponential integral. This implies e.g...
\mathbb {C} } and z ∈ Ω a {\displaystyle z\in \Omega _{a}} , an asymptoticexpansion of Φ ( z , s , a ) {\displaystyle \Phi (z,s,a)} for large a {\displaystyle...
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...
_{-\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 asymptoticexpansion of ψ {\displaystyle \psi } . ψ ( z ) = log z − 1 2 z − ∫ 0 ∞ (...
found in the article on the statistics of random permutations. An asymptoticexpansion for the number of derangements in terms of Bell numbers is as follows:...
digit zero should not be used. Asymptoticexpansion: Approximation of functions generalizing Taylor's formula Asymptotically optimal algorithm: A phrase...
described by an equation involving Big O notation (see also asymptoticexpansion). The series expansion on an open interval will also be an approximation for...
identity reduces to 1 = 1 , {\displaystyle 1=1,} so we must look at its asymptoticexpansion. The Maclaurin series for the cosine function can be written as cos...
looking at the asymptoticexpansion of an analytic function. Since an analytic function is continuous you would expect the asymptoticexpansion to be continuous...
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...
series the error is distributed along the domain of the function. Asymptoticexpansion Generating function Laurent series Madhava series Newton's divided...
iteration of the procedure may generate additional terms of an asymptoticexpansion providing a more accurate solution. An early example of the dominant...
> 0; and that the expansion of f can be term wise differentiated. Then, Laplace–Erdelyi theorem states that the asymptoticexpansion of the integral I(λ)...
2307/2946540, JSTOR 2946540, S2CID 12699956. Erdelyi, A. (1956), AsymptoticExpansions, Dover. Fedoryuk, M. V. (2001) [1994], "Saddle point method", Encyclopedia...
Stirling's approximation, or equivalently the log-gamma function's asymptoticexpansion, log ( k n n , n , ⋯ , n ) = k n log ( k ) + 1 2 ( log ( k...
\right]} where ! ! {\displaystyle !!} denotes the double factorial. An asymptoticexpansion of the cumulative distribution function for large x can also be derived...