Download Asymptotics and Mellin-Barnes Integrals by R. B. Paris PDF
By R. B. Paris
Asymptotics and Mellin-Barnes Integrals presents an account of the use and homes of one of those complicated necessary illustration that arises usually within the learn of certain services regularly of curiosity in classical research and mathematical physics. After constructing the houses of those integrals, their use in settling on the asymptotic habit of targeted capabilities is unique. even if such integrals have an extended heritage, the book's account contains fresh learn leads to analytic quantity idea and hyperasymptotics. The e-book additionally fills a spot within the literature on asymptotic research and certain capabilities through offering a radical account of using Mellin-Barnes integrals that's another way now not on hand in normal references on asymptotics.
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Additional resources for Asymptotics and Mellin-Barnes Integrals
Should an an vanish, the corresponding en is taken to be the zero function, so that the corresponding term in the sum defining f is effectively excised. 13) since |1−e−ζ | ≤ |ζ | when | arg ζ | ≤ 21 π. The series defining f therefore converges uniformly on compact subsets of our sector, and so defines a holomorphic function there. 14) where it bears noting that the infinite series here is uniformly convergent. Because of the exponential decay of each term in the finite sum on the right, the entire sum is o(z−n ) for any n as z → ∞ in our sector.
Steepest descent method. An integral of the form I (λ) = C g(z)eλf (z) dz, is said to be of Laplace type if the functions f and g are holomorphic in a region containing the contour C, and the integral converges for some λ. In the most common setting, C is an infinite contour, and the parameter λ is large in modulus. Thus, we require that the integral I (λ) exist for all λ sufficiently large in some sector. The idea behind the steepest descent method is deceptively simple: deform the integration contour C into a sum of contours, C1 , C2 , .
1) with p = q = 1 and p = 2, q = 1. Example 1. As a first example, let us consider the ratio of two gamma functions (a + bs)/ (1 + s), where a is arbitrary and b > 0. 27) (κs + ϑ + j ) (κs + ϑ + M) (1 + s) j =0 where σM (s) = O(1) as |s| → ∞ uniformly in | arg s| ≤ π − , parameters are given by κ = 1 − b (> 0), ϑ = a − 21 , ϑ = 3 2 > 0. The − a, h = bb and A0 = (2π ) 2 κ 2 −ϑ bϑ . 27) but with κ now given by κ = b − 1 > 0 and h = b−b . 30) in inverse powers of s. Let us consider the case 0 < b < 1.