Hadamard Expansions and Hyperasymptotic Evaluation: An by R.B. Paris

By R.B. Paris

The writer describes the lately built idea of Hadamard expansions utilized to the high-precision (hyperasymptotic) evaluate of Laplace and Laplace-type integrals. This fresh process builds at the famous asymptotic approach to steepest descents, of which the hole bankruptcy offers an in depth account illustrated by way of a sequence of examples of accelerating complexity. A dialogue of uniformity difficulties linked to numerous coalescence phenomena, the Stokes phenomenon and hyperasymptotics of Laplace-type integrals follows.

The closing chapters care for the Hadamard growth of Laplace integrals, with and with out saddle issues. difficulties of other varieties 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 conceivable. the writer additionally considers purposes to a couple very important specified services. This ebook is perfect for graduate scholars and researchers operating in asymptotics.

Show description

Read Online or Download Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents (Encyclopedia of Mathematics and Its Applications Series, Volume 144) PDF

Similar mathematics books

Mathematics of Complexity and Dynamical Systems

Arithmetic of Complexity and Dynamical structures is an authoritative connection with the fundamental instruments and ideas of complexity, structures idea, and dynamical platforms from the point of view of natural and utilized arithmetic.   complicated platforms are structures that include many interacting elements being able to generate a brand new caliber of collective habit via self-organization, e.

GRE Math Prep Course

Each year scholars pay up to $1000 to check prep businesses to organize for the GMAT. you can now get an identical guidance in a publication. GMAT Prep path presents the identical of a two-month, 50-hour direction. even supposing the GMAT is a tricky try, it's a very learnable try. GMAT Prep direction offers an intensive research of the GMAT and introduces a number of analytic strategies to help you immensely, not just at the GMAT yet in company university besides.

Optimization and Control with Applications

This booklet comprises refereed papers that have been provided on the thirty fourth Workshop of the foreign institution of arithmetic "G. Stampacchia,” the foreign Workshop on Optimization and regulate with purposes. The e-book comprises 28 papers which are grouped in line with 4 vast issues: duality and optimality stipulations, optimization algorithms, optimum regulate, and variational inequality and equilibrium difficulties.

Spaces of neoliberalization: towards a theory of uneven geographical development

In those essays, David Harvey searches for sufficient conceptualizations of area and of asymmetric geographical improvement that may aid to appreciate the recent ancient geography of world capitalism. the idea of asymmetric geographical improvement wishes additional exam: the intense volatility in modern political fiscal fortunes throughout and among areas of the realm financial system cries out for larger historical-geographical research and theoretical interpretation.

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

Sample text

22 Asymptotics of Laplace-type integrals where f (t + ) dt + dt − 1 1 − f (t − ) = u −1/2 (1 + u 1/2 )ν + u −1/2 (1 − u 1/2 )ν du du 2 2 ∞ 1 ν = u (k−1)/2 {1 + (−)k } k 2 k=0 ∞ = k=0 ν 2k 1 (|u| < 1). 19), we obtain the expansion eλ I (λ) ∼ √ π ∞ k=0 ∞ ν 2k 0 1 eλ u k− 2 e−λu du = √ π ∞ k=0 ν 2k (k + 12 ) 1 λk+ 2 . Use of the duplication formula for the gamma function and the result ν! = (−ν)2k = (− 12 ν)k ( 12 − 12 ν)k 22k (ν − 2k)! enables us to express the above expansion in the alternative form I (λ) ∼ eλ ∞ (− 12 ν)k ( 12 − 12 ν)k (λ → +∞).

This expansion has been obtained using the method of stationary phase in Wong (1989, p. 82). 30) where, with c > 0, the integration path is taken parallel 13 to the real axis in the upper half-plane and x is a positive variable (Lauwerier, 1966, p. 52). The phase function ψ(t) has a pole at t = 0 and three saddles where ψ (t) = 0 situated on the unit circle at ts = 1 and ts = e±2πi/3 . The saddles at ts = e±2πi/3 are of unit height (that is, √ e−xψ(ts ) = 1), whereas the saddle at ts = 1 has height exp{−3x(1 + i 3)/2} and so is subdominant for x > 0.

13) are given by ak = ψ (k+2) (ts ) , (k + 2)! In particular, we obtain a0 = b0 = e να 2 /2 1 2 bk = { f (ts ) (ts )}(k) . k! 11), ∞ 2 (iα) = q n e2πin , = ω − αν. 15), the first few coefficients ck(1) are (1) c0 = c2(1) = b0 , (2 cosh α)1/2 (1) c1 = 1 ( 1 ib0 tanh α + b1 ), cosh α 3 2 ( 1 b0 (3 − 5 tanh2 α) + ib1 tanh α + 2b2 ), . . (2 cosh α)3/2 12 (2) For the integral I2 , the corresponding coefficients ck about the saddle point are (1) obtained from the ck by replacing (t) by unity in the coefficients bk .

Download PDF sample

Rated 4.26 of 5 – based on 29 votes