The Plemelj Formula of Higher Order Partial Derivatives of the Bochner-Martinelli Type Integral

Abstract.

In this paper, by using the technique of integral transformation, we obtain the Plemelj formulas with the Cauchy principal value and the Hadamard principal value of mixed higher order partial derivatives for integral of the Bochner-Martinelli type on a closed smooth manifold ∂D in Cⁿ. From the Plemelj formulas and using the theory of complex partial differential equation, we prove that the problem of higher order boundary value D^κΦ⁺(t) = D^κΦ⁻(t) + f(t) is equivalent to a complex linear higher order partial differential equation. Moreover, given a proper condition of the Cauchy boundary value problem, the problem of higher order boundary value has a unique branch complex harmonic solution satisfying Φ⁻(∞) = 0 in Cⁿ\∂D.