A simple extension of Fourierʼs integral theorem and some physical applications, in particular to the theory of quanta

(1)Introductory .—In Poincaré’s proof of the necessity of Planck’s hypothesis of quanta, an essential stage of the argument depends on the use of Fourier’s integral theorem to invert a particular infinite integral. In the form used by Poincaré this theorem may be enunciated thus:—Under suitable conditions, if Φ(α ) = ∫∞ 0 e –αη w (η )dη , (1)then w (η ) = 1/2πi ∫c Φ(α )e αη dη , (2)where c is a contour in the complex α-plane on which R(α ) >γ > 0 and I(α )goes from -i ∞ to +i ∞. Poincaré develops an argument which shows that, ifw (η )dη is thea priori probability that the energy of a resonating electron lies betweenη andη +dη , then Φ(α ) is such that -d {log Φ(α )}/dα is the mean energy of the resonator at an absolute temperature C/α , where C is a known constant. When the mean energy of the resonator (of frequencyv ) is known by experiment as a function of the absolute temperature, then Φ(α ) is known, except for an arbitrary constant multiplier. A direct appeal to formula (2) then shows that in these conditions, and with the same exception,w (η ) is also known and is, in fact, unique. It follows at once, and this is the object of Poincaré’s work, that the known facts can be accounted for by one, and only one, function,w (η )—that is, in short, by the hypothesis of quanta.