Partial Differential Equations in Several Complex Variables

2001 ◽  
Author(s):  
So-Chin Chen ◽  
Mei-Chi Shaw
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Partial Differential

scholarly journals Decomposition theorems for the generalized metaharmonic equation in several independent variables

1972 ◽  
Vol 13 (1) ◽  
pp. 35-46
Author(s):  
David Colton
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Radiation Condition ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Entire Space ◽  
Partial Differential ◽  
Independent Variables ◽  
Image Position ◽  
Decomposition Theorems

In this paper solutions of the generalized metaharmonic equation in several independent variables where λ > 0 are uniquely decomposed into the sum of a solution regular in the entire space and one satisfying a generalized Sommerfeld radiation condition. Due to the singular nature of the partial differential equation under investigation it is shown that the radiation condition in general must hold uniformly in a domain lying in the space of several complex variables. This result indicates that function theoretic methods are not only the correct and natural avenue of approach in the study of singular ordinary differential equations, but are basic in the investigation of singular partial differential equations as well.

scholarly journals Regular functions with values in a noncommutative algebra using Clifford analysis

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1747-1755
Author(s):  
Su Lim ◽  
Kwang Shon
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Clifford Analysis ◽  
Regular Function ◽  
Complex Variables ◽  
Partial Differential ◽  
Noncommutative Algebra ◽  
Regular Functions ◽  
Commutative Subalgebra ◽  
Pauli Matrices

We construct a noncommutative algebra C(2) that is a subalgebra of the Pauli matrices of M(2;C), and investigate the properties of solutions with values in C(2) of the inhomogeneous Cauchy-Riemann system of partial differential equations with coefficients in the associated Pauli matrices. In addition, we construct a commutative subalgebra C(4) of M(4;C), obtain some properties of biregular functions with values in C(2) on in C2 x C2, define a J-regular function of four complex variables with values in C(4), and examine some properties of J-regular functions of 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 ◽  
Image Position

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.

Appell Hypergeometric Functions

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Monodromy Group ◽  
Complex Variables ◽  
Partial Differential ◽  
Line Arrangement ◽  
Monodromy Groups ◽  
Complete Quadrilateral

This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the Appell hypergeometric function. Appell introduced four series F1, F2, F3, F4 in two complex variables, each of which generalizes the classical Gauss hypergeometric series and satisfies its own system of two linear second order partial differential equations. The solution spaces of the systems corresponding to the series F2, F3, F4 all have dimension 4, whereas that of the system corresponding to the series F1 has dimension 3. This chapter focuses on the F1-system whose monodromy group, under certain conditions, acts on the complex 2-ball. It first considers the action of S5 on the blown-up projective plane before turning to Appell hypergeometric functions, arithmetic monodromy groups, and an invariant known as the signature.

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