# Bayes, Boltzmann and Bohm - Probabilities in Physics by J. Bricmont

By J. Bricmont

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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 → ∞.