I am trying to read the following paper by Leon Brillouin (the part on page 16 onwards):
Léon Brillouin, Sur une méthode de calcul approchée de certaines intégrales dite méthode du col, Annales scientifiques de l’É.N.S. $3^\text{e}$ série, tome 33 (1916), pp 17–69, doi:10.24033/asens.690.
Is there some English translation to this paper?
I tried using Google translate, but it makes a poor job. Thanks!
The poor quality of Google translate seems to be entirely due to the poor quality of the OCR in the linked pdf file. I cleaned up the OCR and then the first two pages are translated as follows by Google, without any post-editing of the text (I only removed the display equations and added $ signs for the inline formulas). I would think the translation is clear, with the exception of the technical term "neck method", which we would read as "saddle point method".
I want to give some applications of a very fertile approximation method, called the neck method; it makes it possible to obtain approximate formulas for integrals relating to trigonometric or exponential functions; such integrals occur in the wave theories and in all the problems that are treated by means of the Fourier integrals. I will study the following types:
These various integrals occur in the theory of diffraction of light.
The integral of Airy gives the fringes near a caustic: $A (\nu)$ corresponds to the case of a practically unlimited opening; $A (\nu, r, s)$ in the case of a rectangular diaphragm ($^1$). Consider a wave produced by an optical system affected by aberrations. We assume a rectangular diaphragm; we can then, decomposing the wave into spindles, replace the wave by its equator $EE'$.
Let $C$ be a point of the caustic $CC'$ and $OC$ the radius tangent to the caustic in $C$.
The difference of a point $M$ of the wave at point $C$ is of the form
By asking
relative to a point $Q$ located on the normal to the caustic in $C$, the difference of the steps is easy to calculate, if one supposes $Q$ neighbor of the caustic and the small opening (that is to say $M$ neighbor of 0). We find
The phase difference is
and here is the last page
I do not need to dwell on the practical applications of these formulas. They make it possible to study completely the question of the separating power of optical instruments for which the aberration of sphericity exists. Lord Rayleigh had already remarked on this subject that it is not enough to know the interferences only in the focal plane of the central rays. As a result of the interferences, it is not in this plane that the central light spot is the most narrow, but in an intermediate plane between the focus of the central rays and that of the marginal foci. It is on this level that one automatically develops when setting the instrument.
In some series of experiments on star scintillation, the displacement of the interference fringes at the focus is essentially observed. Having put the star in focus, the eyepiece is pushed in slightly so that the image looks like a circle of light. As a result of scintillations, this circle widens or narrows. The variation in appearance of this image gives some information on scintillation. Our precise formulas, allowing to calculate exactly the position of the interference fringes near the home, will make it possible to draw precise information of these experiments.
In the vast majority of cases, it will be possible to simplify the use of these formulas; very often it will suffice to keep only the first term of development, sometimes two or three terms. It would only be for very high precision experiments that complete developments should be used.
The whole text is 54 pages, so this is just 6%, but it only took me five minutes, so I imagine this is entirely doable if there is sufficient interest.
