Representation formulas for the general derivatives of the fundamental solution to the Cauchy-Riemann operator in Clifford analysis and applications
The authors discusss several essentially different formulas for the general derivatives \(q_n(z)\) of the fundamental solution of the Cauchy-Riemann operator in Clifford analysis. Using Fourier and plane wave decomposition methods, they obtain a compact integral representation formula over a half-space which also lends itself to establish upper bound on the values \(\|q_n(z)\|\). Then they further prove several formulas for \(q_n(z)\) in terms of explicit, non-recurrent finite sums, leading themselves to further representations in fewer hypercomplex variable than used in the recurrence relations. Finally, they apply one of the closed representation formulas for \(q_n(z)\) to express the variant of the Riemann zeta function as a finite sum of real-valued Dirichlet series.
scientific article; zbMATH DE number 1827599
scientific article