Fourier, Cauchy and Poisson, working during the same period of time, and – so it would appear – independently, introduced the integral transform which has, over the years, proved itself to be one of the most powerful instruments to gain new results in the field of analysis and its applications. A number of proofs of the convergence of the corresponding integral formula were adduced by Cauchy and Poisson, whereas Fourier, to whom paternity of the formula is ascribed as a matter of course, only presented the single one. Yet another proof was arrived at by a little known mathematician, Camille Deflers. The paper proceeds with a comparative survey of the various proofs propounded in the 1810s and 1820s, leading to a suggested ification of such proofs, on the basis of the method involved. One may thus distinguish those proofs using the auxiliary factor technique (Cauchy, Poisson), those relying on an “evaluation of the weight of the integral” (Deflers, Fourier, Poisson), and finally that presented by Cauchy, bringing in the calculus of residues.