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.
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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 .