Advanced search

Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications

Author
Organization
Abstract
In this paper, we discuss several essentially different formulas for the general derivatives q(n)(z) of the fundamental solution of the Cauchy-Riemann operator in Clifford Analysis, upon,which - among other important applications - the theory of monogenic Eisenstein series is based. Using Fourier and plane wave decomposition methods, we obtain a compact integral representation formula over a half-space, which also lends itself to establish upper bounds on the values parallel toq(n)(z)parallel to. A second formula that we discuss is a recurrence formula involving permutational products of hypercomplex variables by which these estimates can be obtained immediately. We further prove several formulas for q(n)(z) in terms of explicit, non-recurrent finite sums, leading themselves to further representations in terms of permutational products but using different and fewer hypercomplex variables than used in the recurrence relations. Summing up a fixed q(n). over a given discrete lattice leads to a variant of the Riemann zeta function. We apply one of the closed representation formulas for q(n)(z) to express this variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series.
Keywords
fundamental solution, Cauchy-Riemann operator, permutational products, hypercomplex variables, Dirichlet series

Citation

Please use this url to cite or link to this publication:

Chicago
Constales, Denis, and Rolf Krausshar. 2002. “Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and Applications.” Zeitschrift Fur Analysis Und Ihre Anwendungen 21 (3): 579–597.
APA
Constales, D., & Krausshar, R. (2002). Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications. ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN, 21(3), 579–597.
Vancouver
1.
Constales D, Krausshar R. Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications. ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN. 2002;21(3):579–97.
MLA
Constales, Denis, and Rolf Krausshar. “Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and Applications.” ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN 21.3 (2002): 579–597. Print.
@article{161530,
  abstract     = {In this paper, we discuss several essentially different formulas for the general derivatives q(n)(z) of the fundamental solution of the Cauchy-Riemann operator in Clifford Analysis, upon,which - among other important applications - the theory of monogenic Eisenstein series is based. Using Fourier and plane wave decomposition methods, we obtain a compact integral representation formula over a half-space, which also lends itself to establish upper bounds on the values parallel toq(n)(z)parallel to. A second formula that we discuss is a recurrence formula involving permutational products of hypercomplex variables by which these estimates can be obtained immediately. We further prove several formulas for q(n)(z) in terms of explicit, non-recurrent finite sums, leading themselves to further representations in terms of permutational products but using different and fewer hypercomplex variables than used in the recurrence relations. Summing up a fixed q(n). over a given discrete lattice leads to a variant of the Riemann zeta function. We apply one of the closed representation formulas for q(n)(z) to express this variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series.},
  author       = {Constales, Denis and Krausshar, Rolf},
  issn         = {0232-2064},
  journal      = {ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN},
  keyword      = {fundamental solution,Cauchy-Riemann operator,permutational products,hypercomplex variables,Dirichlet series},
  language     = {eng},
  number       = {3},
  pages        = {579--597},
  title        = {Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications},
  volume       = {21},
  year         = {2002},
}

Web of Science
Times cited: