ABSTRACT
Luigi Fantappiè found a new approach to the Cauchy problem for partial differential equations based on some ideas of functional and complex analysis. To solve a Cauchy problem means in terms of functional analysis to construct a linear operator which maps a data to the solution. Fantappiè proposed a lot of analytic methods to represent such a mixed functional in quadratures. His methods start from the notion of indicatrice of an analytic functional. This chapter derives a well known representation of the delta-function as a superposition of plane wave in the form of holomorphic waves. It also discusses the application of holomorphic waves to partial differential equations.
