This monograph, an unaltered reprint of the 1941 edition, is a thorough look at the analytical properties of the Laplace transform. The book develops most of the background needed; it assumes a moderate knowledge of real and complex analysis. The book is aimed at the graduate-student level, but there are no exercises or examples.

The back-cover blurb says the book is “highly theoretical in its emphasis”, which I think is a code-phrase for “no applications”. The book deals primarily with the Laplace transform in isolation, although it does include some applications to other parts of analysis and to number theory. Everything is handled in terms of the (Riemann–)Stieltjes integral, in order to give a unified treatment that covers both integral transforms and generalized Dirichlet series. The book derives all needed properties of these integrals.

Most of the book is aimed at the representation problem (which functions can be expressed as a Laplace transform) and the inversion problem (how to recover a function from its Laplace transform). Leading up to this is a careful study of the Hausdorff moment problem. There is also a long chapter on Tauberian theorems for Laplace transforms, that is not closely related to the rest of the book but is where its applications are. These include the prime number theorem, Littlewood’s strengthening of the original Tauber theorem, and a proof of Wiener’s General Tauberian Theorem.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.