On Laplace ' S Integral Equations
نویسنده
چکیده
which is known in the literature as Laplace's integral equation. The contour (C) and the function f(z) are supposed given and F(x) is to be found. In the case when the contour (C) consists of the positive part of the axis of reals, the solution of the equation (*■) was given by H. Poincaré t and H. Hamburger. % Each of these authors considers F(x) as a function of the real variable *. When the contour (C) consists of the entire axis of reals, a simple substitution reduces (•) to the form studied by Riemann§ and H. Mellin.|| In the present paper we discuss the equation (*) in the case of Poincaré, extending the solution F(x) to complex values of x. A certain relation of reciprocity between the functions f(z) and F(x) is thereby revealed. 1. Poincaré obtained the solution of the equation
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