In 1922 Norbert Wiener, treating the Brownian motion of a particle, introduced a measure on the space of continuous real functions, and a corresponding integral. In 1948 Richard Fcynman, studying the quantum mechanics of a particle, introduced a different integral over the same space. He also showed that his integral can be used to represent the solution of the initial value problem for the Schrodinger equation. This suggested that the Wiener integral can likewise be used to represent the solution of the initial value problem for the heat equation, and Mark Kac showed this in 1949. Since then function space integrals have been used often in physics and studied extensively in mathematics.
We shall present an introduction to the Feynman integral, beginning with a heuristic definition of it in section 1. Then in section 2 we shall show that it solves the Schrodinger equation, and we shall define it for regions with boundaries in section 3. In section 4 we shall define it precisely. In the remaining sections we shall illustrate its use by evaluating it asymptotically. Our purpose is to show how function space integrals can be used to solve partial differential equations, and also how the application of mathematics has again led to the development of a new branch of mathematics.
Further information about the Feynman and Wiener integrals is contained in references and respectively.