On D. J. Patil's remarkable generalisation of Cauchy's formula

Summary

According to a classical theorem of F. and M. Riesz, the values in the open unit disc of an analytic function f (z) of the Hardy class H^p are uniquely determined by the boundary values on a subset E of positive linear measure of the circumference. Patil's formula gives these values explicitly, and Cauchy's formula is the special case where the subset E reduces to the circumference itself. More recently Patil has extended his formula to functions of several complex variables on a poly disc. Patil has obtained these results by functional analysis, using operators and Toeplitz matrices. In the case of one complex variable, an elementary proof is possible, in fact it is at once suggested by the classical devices of Phragmen-Lindelöf, at least in the case where the subset in question consists of a finite sum of arcs. A simple way of passing from this special case to the general case has been given by Steve Wainger. The applications of these methods, and of the related methods of Albert Baernstein, are far-reaching.

One naturally thinks of the Goldbach problem: the Hardy-Little-wood attack amounts to estimating at the origin the n^th derivative of a polynomial of degree 2n, which is well-behaved on most of the unit circumference (on the ‘major arcs’). However I shall limit myself to the case in which E reduces to a single arc. The formula in this case was almost obtained by Paley-Wiener. It provides, among other things, an automatic method of analytic continuation, of the same power as the Lindelöf method of summation of a divergent series.