By R.B. Paris

ISBN-10: 1107096138

ISBN-13: 9781107096134

The writer describes the lately built thought of Hadamard expansions utilized to the high-precision (hyperasymptotic) review of Laplace and Laplace-type integrals. This fresh process builds at the recognized asymptotic approach to steepest descents, of which the outlet bankruptcy offers a close account illustrated via a sequence of examples of accelerating complexity. A dialogue of uniformity difficulties linked to a variety of coalescence phenomena, the Stokes phenomenon and hyperasymptotics of Laplace-type integrals follows.

The last chapters care for the Hadamard enlargement of Laplace integrals, with and with out saddle issues. difficulties of alternative different types of saddle coalescence also are mentioned. The textual content is illustrated with many numerical examples, which aid the reader to appreciate the extent of accuracy possible. the writer additionally considers functions to a few vital precise services. This e-book is perfect for graduate scholars and researchers operating in asymptotics.

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Extra resources for Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents (Encyclopedia of Mathematics and Its Applications Series, Volume 144)

Example text

K! 29), where E p (x) = e0 (e; p); that is, E p (x) = X 0ϑ X 0 e p ∞ A2k X 0−k (2 < p < 6). k=1 For example, when p = 3, we have κ = 23 , ϑ = − 16 ; then, we find ∞ H3 (x) = k=0 i k+1 (3k + 1) k! 25), E 3 (x) = 13 X −1/6 e X e 3πi/4 ∞ A2k X −k e−(6k+1)πi/8 , k=0 where X = 2(x/3)3/2 . When p = 6, a second saddle t1 crosses the positive imaginary t-axis with the consequence that, when 6 < p < 10 (N = 2), there are now two exponentially small contributions which result from these saddles. In this case, we have E p (x) = e0 (x; p) + e1 (x; p), so that E p (x) = X 0ϑ X 0 e p ∞ k=0 A2k X 0−k − X 1ϑ X 1 e p ∞ k=0 A2k X 1−k (6 < p < 10).

9. 9 The graph of 15 eπ ν/2 K 20 iν (x) 25 when ν = 15. was concerned with the representation of K iν (x) in terms of non-oscillatory integrals suitable for numerical quadrature. We write K iν (x) = 1 2 ∞ −∞ e−xψ(t) dt, ψ(t) = cosh t − iνt/x. 1) The saddle points are given by ψ (t) = 0, that is the solutions of the equation sinh t = iν/x. We define the new variable u = ψ(t) − ψ(ts ), so that in the case of a simple saddle point ts u = cosh ts (t − ts )3 (t − ts )4 (t − ts )2 + tanh ts + + ··· .

18) is given by 2 −3/2 k 3π k/2 3u 4 (k → ∞). 19) converge in the disc |u| < 43 . 19), Ai(z) = ∼ z 1/2 −2λ/3 e 2πi z 1/2 −2λ/3 e 2π ∞ 0 ∞ k=0 e−λu dt − dt + − du du du (−)k (3k + 12 ) . λk+1/2 Upon substituting the value λ = z 3/2 , we finally obtain the desired asymptotic expansion of Ai(z) given by Ai(z) ∼ z −1/4 exp(− 23 z 3/2 ) 2π ∞ k=0 (−)k (3k + 12 ) 32k (2k)! 21) as z → +∞. 16) as exp{|λ|ψ(t)}, where now ψ(t) = e3iθ/2 ( 13 t 3 − t). Then it is easily seen that the directions of the steepest descent paths at the saddle point ts = 1 are ± 12 π − 34 θ, whereas those at the saddle point ts = −1 are − 34 θ and π − 34 θ.

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Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents (Encyclopedia of Mathematics and Its Applications Series, Volume 144) by R.B. Paris


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