Received July 26, 1939; in revised form, January 22, 1941. This paper is based, in part, upon the author's dissertation written at Brown University under the direction of Professor J. D. Tamarkin.
Introduction
The Laplace integral
∞ ∫ 0 |
e^{–sx}F(x) dx, |
where x is real and s is real or complex, has been the subject of many extensive investigations. More recently, the Laplace-Stieltjes integral
∞ ∫ 0 |
e^{–sx}F(x) dφ(x) |
has been studied (see, for example, the series of papers by D. V. Widder which appears in the Transactions of the American Mathematical Society, vols. 31, 33, 36, 39); this integral includes as special cases the ordinary Laplace integral and the Dirichlet series.
The object of this paper is to investigate the analogous integral for functions of two variables:
∞ ∫ 0 |
∞ ∫ 0 |
e^{–sx–ty}d_{x}d_{y}φ(x,y). |
in some cases, the results are direct generalizations of the one variable theory; in others, the methods and results are quite different.