Topics on hyperbolic function theory in geometric algebra with a positive signature
Let \(\text{Cl}_k\) be real Clifford algebras. The authors introduce the notion of \(k\)-hypergenic \(\text{Cl}_{n+1}\)-functions (left and right versions). These functions are the null-solutions of the modified Dirac operator \[ H^\ell_k u:= D_\ell u-{k\over x_0} Q_0 u. \] \(Q_0\) is defined by the decomposition \(\text{Cl}_{n+1}= \text{Cl}_n\oplus e_0\text{Cl}_n\). This operator is the projection on the second part. A corresponding modified Laplace-Beltrami operator is studied. It can be shown that a \(\text{Cl}_{n+1}\)-valued function is \(k\)-hypergenic iff \(Ef\) is \(k\)-hypergenic. \(E\) is the Euler operator. For \(k\)-hypergenic functions a Cauchy-type integral-formula is obtained. Using the multivector calculus, interesting new representations for \(H^\ell_{n-1}\) are obtained.
scientific article; zbMATH DE number 5723311