Application of the fundamental principle to complex Cauchy problem (Q1426912)

The non-homogeneous complex Cauchy problem \[ P\biggl({\partial\over \partial z}\biggr) f=g\quad \text{in}\;\Omega, \] \[ f(0,z')=h_0(z'), \ldots {\partial^{m-1}f\over \partial z_1^{m-1}}(0,z')=h_{m-1}(z') \] is considered in a bounded smooth strictly convex domain \(\Omega\subset {\mathbb C}^n,\) \(n\geq 2.\) Here \(g\) is a holomorphic function in a neighbourhood of \(\overline\Omega\), and \(P\in \mathbb C[z_1,\ldots,z_n]\) is a polynomial of degree \(m\) of the form \(P=\sum_{| \alpha| \leq m}a_\alpha z^\alpha.\) It is given an explicit formula for the solution \(f\) obtained using the explicit version of the fundamental principle given in terms of residue currents. Moreover, the domain of definition is characterized and the used techniques are generalized to the non-homogeneous Goursat problem.