By Anton Savin, Boris Sternin

The current monograph is dedicated to the advanced idea of differential equations. no longer but a instruction manual, neither an easy number of articles, the booklet is a primary try to current a kind of certain exposition of a tender yet promising department of arithmetic, that's, the advanced thought of partial differential equations. allow us to attempt to describe the framework of this conception. First, uncomplicated examples convey that suggestions of differential equations are, ordinarily, ramifying analytic features. and, for this reason, usually are not commonplace close to issues in their ramification. moment, taking into account those vital houses of options, we will attempt to describe the strategy fixing our challenge. absolutely, one has first to contemplate differential equations with consistent coefficients. The equipment fixing such difficulties is recognized within the genuine the ory of differential equations: this is often the Fourier transformation. Un thankfully, this sort of transformation had now not but been developed for complex-analytic services and the authors needed to build through them selves. this change is, after all, the most important suggestion of the full theory.

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**Extra resources for Differential Equations on Complex Manifolds**

**Example text**

The first concept which should be mentioned in this context is the Schwarz symmetry principle for solutions of the Laplace equation. The statement of this principle is as follows. Let u(x) be a harmonic function in a domain D c R 2 and let l be an analytic curve in R 2 dividing this domain into two parts D 1 and D 2 (see Fig. 5). toR). The Schwarz symmetry principle is, of course, the first tool for constructing continuations of harmonic functions (that is, of solutions to the Laplace equation).

M - 2, - po=O am-lG(po,p) 8pm I po=O o _1_ H (p)' m where Hrn(P) is the principal symbol of the operator H (- tx). Pod>.. p)' C(p) H being the total symbol of the operator H (- :x) , where the contour C (p) surrounds all zeroes of the denominator in the integrand. 20) can be represented in the form [H (p d~o)] - 1 f= I Po G(po -17 + Po(p*),p*) f (1J,p) d1], po(p*) where Po = Po(P) is the equation of CX. 1. 19)). 13): u(x) = ( . 3. Differential Equations with Constant Coefficients where H(x) is a relative homology class which can be written out explicitly.

Cadilhac [159]-[161], R. Millar investigated the conditions under which Rayleigh's hypothesis can be used (see [134], [135]). His investigation is based also on locating singularities of the continuation of the wave field to domains bounded by the surface of the grating. For example, for two-dimensional scattering problems the Rayleigh's hypothesis amounts to the assumption that the wave field outside the scatterer can be expressed as a series of cylindrical wave functions centered inside the grating.