Introduction. In this paper the problems of Goursat and Cauchy are investigated for equations of hyperbolic type with singular coefficients. Effective algorithms are presented for constructing Duhamel functions of such singular initial problems, reducing these problems to a recurrent sequence of ordinary linear nonhomogeneous differential equations of the second order. Using established differential properties of the operators of Kummer and Gauss, explicit formulas for inverting them are obtained~ as well as simple estimates characterizing the behavior, in the neighborhood of a singular curve of the equation, of the iterations obtained. Studied also are theorems of uniqueness and extremal properties of the solutions of the singular Goursat and Cauchy problems. w i. By the singular Goursat problem we shall understand the problem of finding, in the rectangle = OPMQO, with ver t ices at O(0, 0), P(x0, 0), M(x0, Y0), Q(0, y0), a solution z(x, y) E C2(~) of the equation: L [z, A, B, C] = xz~y + A (x) z~ + B (x) z~ + C (x) z = O, which assumes along the lines x= 0, y=0 , the values (1.1)