Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications
- Author
- Denis Constales (UGent) and Rolf Krausshar
- 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: http://hdl.handle.net/1854/LU-161530
- 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}, }