Reduction of the third boundary value problem by means of harmonic functions
— a(s)<t> = h(s) on 3D. (lb) Here <£„ denotes the normal derivative of 0 in an outward direction. In [1] the theory of analytic functions of a complex variable was used to reduce (1) to a first boundary value problem. Since an analytic function of a complex variable is equivalent to two conjugate harmonic functions, it seems plausible that the method of [1] could be carried out using only harmonic functions and remaining completely in the domain of real numbers. To do this in general is not as easy as it might seem primarily because the separation of the solution obtained there into real and imaginary parts is somewhat involved. In the present paper, we shall indicate some special cases in which it is possible to instrument the ideas in [1] using only harmonic functions. In the last section we also mention a generalization of the method of complex variables to more general types of boundary value problems. 2. Reduction to a first boundary value problem. Under conditions to be formulated below, (1) can be reduced to a first boundary value problem as follows. First find a nonconstant harmonic function, u(x, y), which is constant on 3D. (u may have singularities.) Let v(x, y) be the conjugate of u. Then in a neighborhood of 3D, at least,