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

(2002) ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN. 21(3). p.579-597- 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.

Please use this url to cite or link to this publication:
http://hdl.handle.net/1854/LU-161530

- author
- Denis Constales UGent and Rolf Krausshar
- organization
- year
- 2002
- type
- journalArticle (original)
- publication status
- published
- subject
- keyword
- fundamental solution, Cauchy-Riemann operator, permutational products, hypercomplex variables, Dirichlet series
- journal title
- ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN
- Z. Anal. ihre. Anwend.
- volume
- 21
- issue
- 3
- pages
- 579-597 pages
- Web of Science type
- Article
- Web of Science id
- 000178105100003
- JCR category
- MATHEMATICS
- JCR impact factor
- 0.26 (2002)
- JCR rank
- 136/167 (2002)
- JCR quartile
- 4 (2002)
- ISSN
- 0232-2064
- language
- English
- UGent publication?
- yes
- classification
- A1
- id
- 161530
- handle
- http://hdl.handle.net/1854/LU-161530
- date created
- 2004-01-14 13:39:00
- date last changed
- 2016-12-19 15:38:41

@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}, }

- 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.