On integral operators of bergman type for systems of linear partial differential equations in two complex variables

1992 ◽  
Vol 18 (1-2) ◽  
pp. 103-108
Author(s):  
Peter Berglez
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Operators ◽  
Complex Variables ◽  
Partial Differential ◽  
Linear Partial Differential Equations

Bounds for Solutions of a System of Linear Partial Differential Equations on Domains with Bergman-Silov Boundary

1966 ◽  
Vol 18 ◽  
pp. 1272-1280
Author(s):  
Josephine Mitchell
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Entire Functions ◽  
Integral Operators ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Systems Of Differential Equations ◽  
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The method of integral operators has been used by Bergman and others (4; 6; 7; 10; 12) to obtain many properties of solutions of linear partial differential equations. In the case of equations in two variables with entire coefficients various integral operators have been introduced which transform holomorphic functions of one complex variable into solutions of the equation. This approach has been extended to differential equations in more variables and systems of differential equations. Recently Bergman (6; 4) obtained an integral operator transforming certain combinations of holomorphic functions of two complex variables into functions of four real variables which are the real parts of solutions of the system1where z1, z1*, z2, z2* are independent complex variables and the functions Fj (J = 1, 2) are entire functions of the indicated variables.

scholarly journals The solution of Cauchy's Problem for linear partial differential equations, with constant coefficients, by means of integrals involving complex variables

1928 ◽  
Vol 1 (2) ◽  
pp. 94-103 ◽  
Author(s):  
C. A. Stewart
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Hyperbolic Equation ◽  
Complex Variables ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Constant Coefficients ◽  
Order Hyperbolic Equation ◽  
Second Order Hyperbolic Equation

The object of this paper is to show how the theory of integrals involving complex variables may be applied to the integration of linear partial differential equations, possessing real, distinct characteristics and constant coefficients. The problem considered is a Cauchy problem (with analytic data)—typical of the equation of real characteristics and the method taken is that of Riemann. For simplicity of exposition, the second order hyperbolic equation is considered, but the results are given in such a form as to indicate an obvious generalisation to equations of higher order.

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