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x x 0 displaystyle x to x_ 0 within a sector of validity their difference δ x f x g x displaystyle delta x f x g x is subdominant to both functions a function subdominant in one sector may become dominant in a neighboring sector a process associated with the stokes phenomenon 20 115 116 examples edit plots of the absolute value of the fractional error in the asymptotic expansion of the gamma function left the horizontal axis is the number of terms in the asymptotic expansion blue points are for x 2 displaystyle x 2 and red points are for x 3 displaystyle x 3 it can be seen that the least error is encountered when there are 14 terms for x 2 displaystyle x 2 and 20 terms for x 3 displaystyle x 3 beyond which the error diverges gamma function stirling s approximation 22 e x x x 2 π x γ x 1 1 1 12 x 1 288 x 2 139 51840 x 3 x for arg x π displaystyle frac e x x x sqrt 2 pi x gamma x 1 sim 1 frac 1 12x frac 1 288x 2 frac 139 51840x 3 cdots x to infty text for arg x pi exponential integral 23 x e x e 1 x n 0 1 n n x n x for arg x π displaystyle xe x e_ 1 x sim sum _ n 0 infty frac 1 n n x n x to infty text for arg x pi logarithmic integral for real variable x displaystyle x 23 li x x ln x k 0 k ln x k x displaystyle operatorname li x sim frac x ln x sum _ k 0 infty frac k ln x k x to infty riemann zeta function 24 ζ s n 1 n n s n 1 s s 1 n s 2 n s m 1 b 2 m s 2 m 1 2 m n 2 m 1 for arg s π re s 2 n n 1 2 3 displaystyle zeta s sim sum _ n 1 n n s frac n 1 s s 1 frac n s 2 n s sum _ m 1 infty frac b_ 2m s overline 2m 1 2m n 2m 1 text for arg s pi operatorname re s 2n n 1 2 3 ldots where b 2 m displaystyle b_ 2m are bernoulli numbers and s 2 m 1 displaystyle s overline 2m 1 is a rising factorial this expansion is valid for all complex s displaystyle s and is often used to compute the zeta function by using a large enough value of n displaystyle n for instance n s displaystyle n vert s vert complementary error function 25 π x e x 2 e r f c x 1 n 1 1 n 2 n 1 2 x 2 n x for arg x 3 π 4 displaystyle sqrt pi xe x 2 rm erfc x sim 1 sum _ n 1 infty 1 n frac 2n 1 2x 2 n x to infty text for arg x frac 3 pi 4 where 2 n 1 displaystyle 2n 1 is the double factorial worked example edit this example generates an asymptotic expansion for a function related to the exponential integral defined using the cauchy principal value pv e 1 t ei 1 t e 1 t pv 1 t e u u d u displaystyle e 1 t operatorname ei left frac 1 t right e 1 t operatorname pv int _ frac 1 t infty frac e u u du the substitution w u t 1 displaystyle w ut 1 transforms this expression into e 1 t ei 1 t pv 0 e w t 1 w d w displaystyle e 1 t operatorname ei left frac 1 t right operatorname pv int _ 0 infty frac e w t 1 w dw applying the finite geometric series formula to the integrand s denominator expresses the function as a sum of integrals e 1 t ei 1 t m 0 n 1 0 w m e w t d w pv 0 w n e w t 1 w d w displaystyle e 1 t operatorname ei left frac 1 t right sum _ m 0 n 1 int _ 0 infty w m e w t dw operatorname pv int _ 0 infty frac w n e w t 1 w dw evaluating these integrals generates a finite series with factorial coefficients and an integral remainder e 1 t ei 1 t m 0 n 1 m t m 1 pv 0 w n e w t 1 w d w displaystyle e 1 t operatorname ei left frac 1 t right sum _ m 0 n 1 m t m 1 operatorname pv int _ 0 infty frac w n e w t 1 w dw although the series is divergent truncating the series may provide a good approximation of the exponential integral for sufficiently small t displaystyle t the full asymptotic expansion includes all series terms 26 51 53 e 1 t ei 1 t m 0 m t m 1 t 0 displaystyle e 1 t operatorname ei left frac 1 t right sim sum _ m 0 infty m t m 1 quad t to 0 repeated integration by parts is an alternative method to derive this identical expansion 27 5 substituting x 1 t displaystyle x tfrac 1 t and noting that ei x e 1 x displaystyle operatorname ei x e_ 1 x recovers the expansion given earlier in the article ei x e x x m 0 n m x m e n x e n x n 1 x e x x e t t n 2 d t displaystyle operatorname ei x frac e x x left sum _ m 0 n frac m x m e_ n x right quad e_ n x equiv n 1 xe x int _ infty x frac e t t n 2 dt for any fixed x displaystyle x the absolute value of the error term e n x displaystyle e_ n x decreases then increases as n displaystyle n increases the minimum occurs at n x displaystyle n sim vert x vert at which point e n x 2 π x e x displaystyle textstyle vert e_ n x vert leq sqrt frac 2 pi vert x vert e vert x vert exponentially small truncation errors are common for optimally truncated series 28 27 51 asymptotic expansions may arise when a series is integrated beyond its boundary of convergence 1 111 regularization methods edit optimal truncation edit these regularization methods assign a value to an asymptotic series using truncated series the superasymptotic expansion is the truncated series which minimizes the magnitude of the remainder term a remainder formula or remainder upper bound formula may determine the optimal truncation index n displaystyle n if an exact formula is unavailable heuristic methods may determine the optimum truncation index such as truncating the series immediately before or at its minimum term summation to the least term or using formulas that determine the optimal index n displaystyle n from the value of the asymptotic variable x displaystyle x 29 30 31 32 a term δ x displaystyle delta x is said to lie beyond all orders of an asymptotic expansion if the term is asymptotically smaller than every gauge function φ n x displaystyle varphi _ n x of the expansion δ x o φ n x x l displaystyle delta x o varphi _ n x quad x to l commonly an optimally truncated power series expansion at the origin has a beyond all orders truncation error of order o e q x displaystyle o e q x with positive constant q displaystyle q as x 0 displaystyle x to 0 in the sector arg x π 2 displaystyle arg x pi 2 at the origin this truncation error is non analytic and cannot be expressed as a taylor series 18 1 10 the term asymptotics beyond all orders refers to the collection of methods used to measure the effects of these beyond all order terms 21 viii for an alternating series the remainder is often approximately half of the series least term adding half of the least term to the sum of the preceding terms often generates a highly accurate approximation 1 401 403 the hyperasymptotic expansion replaces the remainder of a superasymptotic expansion with an optimally truncated second series and second smaller remainder often using a different scale repeating this process generates exponentially smaller remainder terms 26 51 summation of the series tail edit these regularization methods compute a series remainder by applying borel summation or mellin barnes regularization to the series tail the expansion s underlying function is the sum of this computed remainder and the truncated series borel summation edit when a truncated series has a factorially divergent tail borel summation evaluates the remainder as the product of the tail s first term and an integral called a terminant for an alternating sign tail m n γ m α 1 x m 0 α 1 arg x π displaystyle sum _ m n infty gamma m alpha 1 x m quad 0 leq alpha 1 arg x pi borel summation generates a type i terminant λ s x displaystyle lambda _ s x defined by a cauchy principal value integral λ s x 1 γ s 1 pv 0 t s e t 1 x t d t displaystyle lambda _ s x frac 1 gamma s 1 operatorname pv int _ 0 infty frac t s e t 1 xt dt multiplying this terminant by the tail s first term generates the exact remainder 33 17 γ n α 1 x n λ n α x displaystyle gamma n alpha 1 x n lambda _ n alpha x for a same sign tail m n γ m α 1 x m 0 α 1 displaystyle sum _ m n infty gamma m alpha 1 x m quad 0 leq alpha 1 the positive real axis forms a stokes line borel summation generates a type ii terminant λ s x displaystyle bar lambda _ s x with the stokes constant s displaystyle s 33 17 λ s x λ s x 2 s i π x s 1 e 1 x γ s 1 s 1 2 if 0 arg x 2 π 0 if arg x 0 1 2 if 2 π arg x 0 displaystyle begin aligned bar lambda _ s x lambda _ s x 2s dfrac i pi x s 1 e 1 x gamma s 1 s begin cases dfrac 1 2 text if 0 arg x 2 pi 0 text if arg x 0 dfrac 1 2 text if 2 pi arg x 0 end cases end aligned the resulting remainder is the product 33 19 γ n α 1 x n λ n α x displaystyle gamma n alpha 1 x n bar lambda _ n alpha x for more general divergent tails a linear combination of late series terms and terminants called a distribution over basic terminants represents the remainder 1 431 515 mellin barnes tail regularization edit mellin barnes regularization evaluates the series tail separately for alternating sign coefficients and same sign coefficients for an alternating sign series tail m n f m x β m displaystyle sum _ m n infty f m x beta m mellin barnes regularization computes this remainder 33 94 c i c i x β s f s e i π s e i π s d s π ε 1 β arg x π ε 2 β displaystyle int _ c i infty c i infty frac x beta s f s e i pi s e i pi s ds quad frac pi varepsilon _ 1 beta arg x frac pi varepsilon _ 2 beta where n 1 c re s n displaystyle n 1 c operatorname re s n 33 94 the asymptotic behavior of the integrand as s displaystyle s approaches the limits of integration determines the constants ε 1 displaystyle varepsilon _ 1 and ε 2 displaystyle varepsilon _ 2 33 84 for a same sign series tail m n f m x β m displaystyle sum _ m n infty f m x beta m the positive real axis forms a stokes line for the sector above the stokes line the method generates this remainder c i c i x β s e i π s f s e i π s e i π s d s 1 2 c i c i x β s f s d s 0 arg x 2 π b β displaystyle int _ c i infty c i infty frac x beta s e i pi s f s e i pi s e i pi s ds frac 1 2 int _ c i infty c i infty x beta s f s ds quad 0 arg x frac 2 pi b beta where n 1 c re s n displaystyle n 1 c operatorname re s n for the sector below the stokes line the remainder becomes c i c i x β s e i π s f s e i π s e i π s d s 1 2 c i c i x β s f s d s 2 π a β arg x 0 displaystyle int _ c i infty c i infty frac x beta s e i pi s f s e i pi s e i pi s ds frac 1 2 int _ c i infty c i infty x beta s f s ds quad frac 2 pi a beta arg x 0 where n 1 c re s n displaystyle n 1 c operatorname re s n 33 101 104 the asymptotic behavior of the integrand as s displaystyle s approaches the limits of integration determines the constants a displaystyle a and b displaystyle b 33 98 99 additional integrals convergent in overlapping sectors allow analytic continuation in the complex plane for both types of series 33 84 113 summation of the full series edit this regularization method applies borel summation or mellin barnes regularization to the entire series computing the borel sum of a typical gevrey order 1 asymptotic series requires a borel transform analytic continuation and a laplace transform the borel transform converts this divergent power series with zero radius of convergence into a convergent series with a nonzero radius of convergence called the borel transformed series 34 a series acceleration method can accurately sum a slow converging series 35 the analytic continuation of the borel transformed series is the borel transformed function the borel transformed function s domain now extends across the complex plane excluding the function s singularities comparing the transformed series coefficients to established tables of function expansions and coefficient sequences may identify the analytic continuation as a special function an integral representation or a generating function 36 47 37 38 39 an alternative approach uses rational function approximation of the transformed series guided by singularity analysis and conformal mapping 40 41 a bijective conformal map uniform map is especially effective for accurate approximation 42 in comparison to the padé approximant the adaptive antoulas anderson aaa algorithm is numerically more stable and less prone to generating spurious poles 43 finally the laplace transform converts the borel transformed function to the borel sum a finite value assigned to the original divergent series 34 convergent alternative expansions edit series such as inverse factorial series or nörlund series binary rational inverse factorial series or modified hadamard series are alternatives to asymptotic expansions because they may converge for functions with divergent asymptotic series 44 the inverse factorial series or nörlund series with coefficients c m displaystyle c_ m has the following form 45 46 f x c 0 m 1 c m x x 1 x 2 x m 1 displaystyle f x c_ 0 sum _ m 1 infty frac c_ m x x 1 x 2 cdots x m 1 a binary rational inverse factorial series uses a double summation over the coefficients c k m displaystyle c_ k m it has the following form 47 f z k 1 m 0 c k m 2 k m m 2 k z 1 m 1 displaystyle f z sum _ k 1 infty sum _ m 0 infty frac c_ k m 2 km frac m 2 k z 1 m 1 this binary rational variant typically converges more rapidly than the standard inverse factorial series 47 a modified hadamard series is a convergent series designed to overcome the divergence or slow convergence of traditional asymptotic expansions each term in the series has a normalized incomplete gamma function that can gradually decrease the magnitude of later terms selecting optimal integration intervals and rearranging how later terms are summed leads to high accuracy 48 100 143 see also edit related fields edit asymptotic analysis singular perturbation asymptotic methods edit watson s lemma mellin transform laplace s method stationary phase approximation method of dominant balance method of steepest descent notes edit 1 2 3 4 5 6 7 dingle 1973 1 2 murray 2012 wong 1989 murray 2012 p 2 verhulst 2006 p 1 paulsen 2013 1 2 estrada kanwal 2012 bayes 1763 barbeau leah 1976 1 2 kowalenko 2011 1 2 berry howls 1991 poincaré 1886 stieltjes 1886 jahnke 2003 van boven wesselink wepster 2012 1 2 olver 1974 de jager furu 1996 1 2 boyd 1999 1 2 3 erdélyi 2012 1 2 3 bender orszag 1978 1 2 segur tanveer levine 1991 olver wong 2010 1 2 temme 2010a apostol 2010 temme 2010b 1 2 boyd 2012 temme 2015 o malley 2014 costin 2008 p 114 boyd 1999 pp 7 8 13 17 temme 2015 p 10 bender orszag 1978 p 121 122 1 2 3 4 5 6 7 8 9 kowalenko 2009 1 2 mera pedersen nikolić 2018 caliceti et al 2007 dunne 2026 sloane 2014 dingle 1973 pp 44 47 56 99 olde daalhuis 2010 dunne 2026 pp 45 52 53 flajolet sedgewick 2009 pp 375 438 costin dunne 2021 trefethen 2023 weniger 2010 temme 2010c olde daalhuis 2004 1 2 costin dunne 2018 paris 2011 references edit ablowitz mark j fokas athanassios s 2003 complex variables introduction and applications cambridge university press isbn 978 1 139 43913 8 aniceto inês başar gökçe schiappa ricardo 2019 a primer on resurgent transseries and their asymptotics physics reports 809 1 135 arxiv 1802 10441 doi 10 1016 j physrep 2019 02 003 issn 0370 1573 apostol t m 2010 zeta and related functions in olver frank w j lozier daniel m boisvert ronald f clark charles w eds nist handbook of mathematical functions cambridge university press isbn 978 0 521 19225 5 mr 2723248 balser werner 2006 from divergent...