Solutions for the generalized Loewner differential equation in several complex variables

2009 ◽  
Vol 347 (2) ◽  
pp. 411-435 ◽  
Author(s):  
Peter Duren ◽  
Ian Graham ◽  
Hidetaka Hamada ◽  
Gabriela Kohr
Keyword(s):  
Differential Equation ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Loewner Differential Equation

scholarly journals On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions

2020 ◽  
Vol 4 (3) ◽  
pp. 33
Author(s):  
Yudhveer Singh ◽  
Vinod Gill ◽  
Jagdev Singh ◽  
Devendra Kumar ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Fractional Order ◽  
Integral Transform ◽  
Special Functions ◽  
Confluent Hypergeometric Function ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Volterra Type

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here.

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 Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

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