Excerpt from The Singularities of the Riemann Function: January 1, 1961

We shall deal with the Cauchy problem for linear hyperbolic systems of the first order, with n+1 independent variables. Our central problem is the description of the singularities of the Riemann function. In addition to its own intrinsic interest, this problem provides a key to the mathematical theory of wave propagation, since any solution of the Cauchy problem can be represented in terms of the Riemann function. In a sense, the singularities of the Riemann function determine the structure of the dependence of a solution on its initial data.

The analysis of the singularities of the Riemann function also represents a step towards an extension of the method of Hadamard to general linear hyperbolic equations. This method would determine the Riemann function by substitution of a function of the proper form into the differential equation.

Our analysis also provides an approach to problems involving propagation of waves in media where the characteristics have variable multiplicity, including questions of existence and uniqueness. Such problems are frequently ignored in the mathematical literature, although they have great physical interest.

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