By Andrew Creese, David Parker

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Ii) If δ > 0, then F = L−1 f is analytic in the sector S = {p = 0 : | arg(p)| < δ}. In addition, supS |F | ≤ K/2 and F (p) → 0 as p → ∞ along rays in S. 17) ids f (is) −∞ i∞ = ∞ dp e−px eips ids eips f (is) = 0 −i∞ where we applied Fubini’s theorem3 and then pushed the contour of integration past x to infinity. The norm is obtained by majorizing |f (x)epx | by K(|x2 | + 1)−1 . 3 This theorem addressesR the permutation of RtheRorder of integration; see [51]. Essentially, R R if f ∈ L1 (A × B), then A×B f = A B f = B A f .

This contour can be deformed, by assumption, to the dotted contour in the figure. 115), since it is smaller than R−k times any power of k, as k → ∞. Now the contribution from each singularity is of the form 1 2π 4A Bl f (s) ds sk+1 convergent series in terms of integer or noninteger powers of the z − zn .

86) −π/2 Suppose we would like to find the behavior of J0 (ξ) as ξ → +∞. 87) and which is easy to analyze by standard ODE means. 88) π/2 as shown in Fig. 1. All the curves involved in this decomposition of I are lines of constant imaginary part of the exponent, and the ordinary Laplace method can be applied to find their asymptotic behavior for ξ → +∞ (note also that the integral along the curve γ, called Sommerfeld contour, is the only one contributing to J0 , the other two being purely imaginary, as it can be checked by making the changes of variable t = −π/2 ± is).