By Ovidiu Costin
Incorporating vast advancements from the final thirty years into one source, Asymptotics and Borel Summability offers a self-contained advent to asymptotic research with designated emphasis on themes no longer coated in conventional asymptotics books. the writer explains uncomplicated principles, strategies, and techniques of generalized Borel summability, transseries, and exponential asymptotics. He presents whole mathematical rigor whereas supplementing it with heuristic fabric and examples, in order that a few proofs can be passed over via applications-oriented readers.
To supply a feeling of the way new tools are utilized in a scientific method, the ebook analyzes intimately basic nonlinear usual differential equations (ODEs) close to a conventional abnormal singular element. It allows readers to grasp easy thoughts, offering an organization origin for extra examine at extra complicated degrees. The e-book additionally examines distinction equations, partial differential equations (PDEs), and different varieties of problems.
Chronicling the development made in fresh many years, this booklet indicates how Borel summability can get well special options from formal expansions, examine singular habit, and greatly enhance accuracy in asymptotic approximations.
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Extra info for Asymptotics and Borel summability
Let f (x) = 0 e ˜ sufficient conditions on F such that f , the asymptotic power series of f for large positive x, is a convergent series for |x| > R > 0. (b) Assume that f˜ converges to f . Show that f is zero. (c) Show that in case (a) if F is analytic in a neighborhood of [0, 1] then f = f˜ + e−x f˜1 where f˜1 is convergent for |x| > R > 0. 41 cannot be extended to more than a half-plane: Show that if f is entire, of exponential order one, and bounded in a 34 Asymptotics and Borel summability sector of opening exceeding π, then it is constant.
Then 2π f (t)eint = o(n−m ) for any m > 0 as n → +∞, n ∈ Z. 0 (2) Assume f ∈ C0∞ [a, b] vanishes at the endpoints together with all derivab tives; then fˆ(x) = a f (t)eixt = o(x−m ) for any m > 0 as x → +∞. 65) have zero radius of convergence. 67 (2) show that lim supx→∞ e for any > 0 unless f = 0. 56 under smoothness assumptions. If we write f (s + π/x) = f (s) + f (s)(π/x) + 1 2 2 f (c)(π/x) cancellation is manifest. 71 Show that if f is piecewise differentiable and the derivative is in L1 , then the Fourier transform is O(x−1 ).
18 (the case when g is maximum at one endpoint). Assume f is continuous on [a, b], f (a) = 0, g is in C 1 [a, b] and g < −α < 0 on [a, b]. 19) Note: Since the derivative of g enters in the final result, regularity is clearly needed. PROOF Without loss of generality, we may assume a = 0, b = 1, f (0) > 0 (the complex case, f = u + iv, is dealt with by separately treating u and v). Classical asymptotics 29 Let be small enough and choose δ such that if s < δ we have |f (s)−f (0)| < and |g (s) − g (0)| < .