# Computational Methods for Physicists: Compendium for by Simon Širca, Martin Horvat

By Simon Širca, Martin Horvat

This e-book is helping complex undergraduate, graduate and postdoctoral scholars of their day-by-day paintings via delivering them a compendium of numerical equipment. the alternative of equipment will pay major recognition to blunders estimates, balance and convergence concerns in addition to to the how one can optimize software execution speeds. Many examples are given in the course of the chapters, and every bankruptcy is via not less than a handful of extra entire difficulties that may be handled, for instance, on a weekly foundation in a one- or two-semester path. In those end-of-chapter difficulties the physics history is mentioned, and the most textual content previous them is meant as an creation or as a later reference. much less rigidity is given to the reason of person algorithms. it's attempted to urge within the reader an personal self sufficient considering and a certain quantity of scepticism and scrutiny rather than blindly following available advertisement instruments.

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For details, see [16]. Padé approximations have numerous important transformation and invariance properties, of which duality and unitarity are the most relevant. Duality connects the Padé approximations for reciprocal functions: g(z) = f (z) −1 and f (0) = 0 ⇔ [L/M]g (z) = [M/L]f (z) −1 ∀L, M. 12 1 Basics of Numerical Analysis In physical applications, unitarity is even more important, in particular in the k theory of scattering matrices. Assume that f (z) = ∞ k=0 ck z is unitary, so that ∗ f (z)f (z) = 1, and that [M/M]f is its diagonal Padé approximation.

2. 3 Power and Asymptotic Expansion, Asymptotic Analysis 27 General integrals along a contour C in the complex plane I (λ) = C g(z) eλf (z) dz, λ → ∞, where f and g are analytic, can be computed by means of the method of steepest descent and by the saddle-point method, which are both similar to the Laplace method in spirit, but technically more complicated. For further information, we refer the reader to [29] and [33]. 6 Differential Equations with Large Parameters Asymptotic approaches are also applicable to the analysis of differential equations.

28) a in the limit of large positive λ, where φ and h are real functions of a real variable x. Assume that h has a global minimum at one of the internal points ξ of the interval [a, b], thus h (ξ ) = 0 and h (ξ ) > 0. 28). 29) while we take simply φ(x) ≈ φ(ξ ). 28) and its integration limits are extended to [−∞, +∞], we obtain b I (λ) ≈ φ(ξ ) e−λ[h(ξ )+h (ξ )(x−ξ )2 /2] dx ≈ φ(ξ ) e−λh(ξ ) a ∞ −∞ e−λh (ξ )x 2 /2 dx. This is the Laplace approximation, which is the leading term in the asymptotic expansion I (λ) = e−λh(ξ ) φ(ξ ) 1 2π +O λh (ξ ) λ , λ → ∞.