Integral representations of solutions of some partial differential equations (Q1382694)

We obtain integral representations of solutions of some differential equations of the type \[ \lambda u_{zz}+ {\partial^2u \over\partial \xi\partial \eta}=0, \quad\xi= x+ iy, \quad \eta= x-iy \tag{1} \] either in terms of holomorphic functions of complex variables or in terms of solutions of other equations. In (1) \(x,y,z\) are real variables, \(u\) is an unknown complex function, \(\lambda\) is a complex constant, and, for a complex \(\lambda\), equation (1) is considered as a system of two real equations. Equation (1) is elliptic for \(\text{Im} \lambda \neq 0\) and for \(\text{Im} \lambda =0\), \(\text{Re} \lambda>0\), while for \(\text{Im} \lambda =0\), \(\text{Re} \lambda <0\) it is hyperbolic. We consider equation (1) in a neighborhood of a domain \(D\) lying in the plane \(z=0\).