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On a new integral transformation of ramifying analytic functions. (English) Zbl 0852.44004
Gliklikh, Yu. E. (ed.), Methods and applications of global analysis. Voronezh: Voronezh University Press. Nov. Global’nom Anal. 101-145 (1993).
This paper is an elementary introduction to the theory of an integral transformation of complex analytic functions. It consists of two parts. The first part is aimed at the consideration of the two-dimensional case. Here we consider first the most simple version of the transformation – the so-called $$\partial/ \partial \xi$$-transformation. From the general point of view it is the technical pattern (which is, however of interest by itself) for constructing of the general (two-dimensional) transformation.
The second part contains the consideration of the case of arbitrary dimensions. Except for the necessary definitions, we formulate here the main theorems on commutation of the introduced transformation with differentiation and multiplication by the independent variable. These theorems are the main ones in applications. Later on, the particular cases of the general transformation are derived from the general formula for the transformation. These particular cases are the $${\mathcal R}$$-transformation and the $$\partial/ \partial \xi$$-transformation which was our starting point in the first part.
For the entire collection see [Zbl 0843.00021].
##### MSC:
 44A15 Special integral transforms (Legendre, Hilbert, etc.) 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations