Get Asymptotic Approximations of Integrals. Computer Science and PDF

By R. Wong

ISBN-10: 0127625356

ISBN-13: 9780127625355

Asymptotic equipment are usually utilized in many branches of either natural and utilized arithmetic, and this vintage textual content is still the main up to date e-book facing one very important point of this region, specifically, asymptotic approximations of integrals. during this publication, all effects are proved carefully, and plenty of of the approximation formulation are observed via errors bounds. an intensive dialogue on multidimensional integrals is given, with references supplied. Asymptotic Approximations of Integrals comprises the 'distributional method', no longer to be had somewhere else. many of the examples during this textual content come from concrete functions. in view that its booklet twelve years in the past, major advancements have happened within the common idea of asymptotic expansions, together with smoothing of the Stokes phenomenon, uniform exponentially enhanced asymptotic expansions, and hyperasymptotics. those new options belong to the realm referred to now as 'exponential asymptotics'. Expositions of those new theories are available papers released in a variety of journals, yet now not but in publication shape

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Example text

Let J(x) = Jo ÍV 0 (JCÍ) di, -1 < α < |, where Jm(t) is the Bessel function of the first kind of order m. Use the identity (tmJm(t))' = tmJm _ ¿t) to show that '<*>= Z^- i r r r 7 è 7J t t ) »-^w V2A fc = l 2 a/ + 2" Π " * j 7 ; g) « - n [r - n¿M) dt. * V2 2a/ Jo This suggests that the integral I(x) may have the generalized asymptotic expansion I(x)~ Σ Ζ * - 1 ^ " 1 ; ^ * - * ^ * ) ; {*"*}> asx-oo. 46 I Fundamental Concepts of Asymptotics Now, use the identity (Erdélyi et ai 1953b, p. 49) rVu^xí) dt = 2τ-*χ-> Γ ^ + ^) Γ Jo ( 1 + \μ~ —Re μ < Re ρ < §, to show that as x -* oo, " [V-V„(xt)dt~x-*-x2 α — loa — n Jo ΪΡ) Γ(| + |α) V¿ ' z / Γ(η + | - | a ) ' thus proving the above generalized asymptotic expansion invalid.

Show by induction that | ff{t)\ < Mkem for i = 0, 1 , . . , 2n - 2k and k = 0, 1 , . . , n. From this deduce that δη(χ) = 0 ( χ - 2 η " Κ β α ) as x -» + oo. 8. Let J(x) = Jo ÍV 0 (JCÍ) di, -1 < α < |, where Jm(t) is the Bessel function of the first kind of order m. Use the identity (tmJm(t))' = tmJm _ ¿t) to show that '<*>= Z^- i r r r 7 è 7J t t ) »-^w V2A fc = l 2 a/ + 2" Π " * j 7 ; g) « - n [r - n¿M) dt. * V2 2a/ Jo This suggests that the integral I(x) may have the generalized asymptotic expansion I(x)~ Σ Ζ * - 1 ^ " 1 ; ^ * - * ^ * ) ; {*"*}> asx-oo.

Show that as n -> oo, Σ> + "' -(1 - 2"> «-> ~ TT^.?. C * > - '"-> ¿ P where £(z) is the Riemann Zeta function. Also show that if the expansion is truncated at the term s = m — 1, where m > |(α + 1), then the remainder is bounded in absolute value by the next term and has the same sign. 22. Let l"-1 (2/ + 1 π Λ„ = - Σ cot — . From the expansion cot« = - - X ^ I f i J - ^ j - , |z|<«, derive the asymptotic series 2 8 m _ * ( — l ) s + xA A„ = - l o g n + A 0 + - X °+ P,», π π s= 1 (¿ra) 53 Exercises where 2rl A0 = - l o g 2 + - y - - X π π π r r ' 1 r(2r)!

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Asymptotic Approximations of Integrals. Computer Science and Scientific Computing by R. Wong


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