In a recent R&D note by Huang et al. (1984), an analytical solution for a packed bed reactor with a first-order reaction is derived. The solution is obtained by the Laplace-transform method and by inversion in the complex plane. Since the transform contains branch points a , so called second Bromwich contour (McLachlan, 1953) is used to obtain the solution in the time domain.Some criticism is also made of an earlier analytical solution of the same problem (but without chemical reaction) given by Rasniuson and Neretnieks (1980). We are supposed to have used the first Bromwich contour (McLachlan, 1953) in our derivation. Since the Laplace transform contains branch points, they contend, this procedure is not valid. This statement is correct. However, in our analytical inversion, neither the first nor second Bromwich contour is used. Instead, we integrate along the imaginary axis. This integration path follows direct application of the complex inversion integral for the Laplace transform. We would like to clarify our analytical inversion of the Laplace transform and discuss relative merits of various methods.Under very general conditions, the inversion integral of the Laplace transform is given (Churchill, 1972; Doetsch, 1976) by:

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These results suggest that the natural width of a spectral line may be less than the value to be expected from the classical theory of damping by radiation.The subject of width of spectral line is one of considerable importance. If the lines are really much narrower than 0.12 X-Unit the radiation cannot come from a damped oscillating electron. The mechanism must be such as to maintain a pure harmonic oscillation of constant amplitude until the quantum of energy is completely emitted. Such a train of waves would need to have a great number of elements and so have considerable length. An alternate hypothesis would be that a quantum is an entity (the word "pulse" is avoided) that may be resolved into a train of waves by the crystal grating. In this case the width of a spectral line would depend on the degree of perfection of the crystal. The quantum theory of crystal grating action advanced by Duane8 might also give a narrow spectral line, whose width would be a property of the crystal grating and not of the radiation. In stu:dying the very significant statistical.interpretation put on the quantum mechanics by the "transformation theory" of Dirac' and Jordan,2 the writer at first experienced considerable difficulty in understanding how the quantum formulas for averages and probabilities merge into the analogous classical expressions in the region of large quantum numbers and also, of course, in the limit h = 0. In the present note we shall aim to trace through the asymptotic connection between the formulas of the two theories, which does not seem to have been quite adequately elucidated in existing papers.In the transformation theory a diagonal element of a matrix which 178 PRtOC.,N. A. S.

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