A generalization of a Cullen’s integral theorem for the quaternions. (English) Zbl 1206.30065
From the introduction: The purpose of this paper is to show how hyperholomorphic functions satisfy the following integral theroem given by C. G. Cullen [Duke Math. J. 32, 139–148 (1965; Zbl 0173.09001)].
Proposition 4. Let \(f=u+\iota v\) be an hyperholomorphic function, and let \(K\) be any smooth, simple, closed hypersurface in \(\mathbb H\), the quaternionic space, disjoint to the real axis, and let \(K^*\) be the interior of \(K\). Let \(n(p)=n_0+n_1i+n_2j+n_3k\), where \((n_0,n_1,n_2,n_3)\) is the unit outer normal to \(K\) at \(p\). Then
\[ \int_K N(p)f(p)\frac{1}{r^2}dS_K =-2\int_{K^*} u\frac{\iota}{r^3} dV, \] where \(dS_K\) is surface element of \(K\).
Proposition 4. Let \(f=u+\iota v\) be an hyperholomorphic function, and let \(K\) be any smooth, simple, closed hypersurface in \(\mathbb H\), the quaternionic space, disjoint to the real axis, and let \(K^*\) be the interior of \(K\). Let \(n(p)=n_0+n_1i+n_2j+n_3k\), where \((n_0,n_1,n_2,n_3)\) is the unit outer normal to \(K\) at \(p\). Then
\[ \int_K N(p)f(p)\frac{1}{r^2}dS_K =-2\int_{K^*} u\frac{\iota}{r^3} dV, \] where \(dS_K\) is surface element of \(K\).
MSC:
| 30G35 | Functions of hypercomplex variables and generalized variables |
