Representation of Complex Integrals and Twisted de Rham Cohomologies
In integral representations of Euler type of classical hypergeometric functions of several variables or of hypergeometric functions which are studied these days, integrals of the product of powers of polynomials appear. We will establish a framework to treat such integrals, and after that, we will study hypergeometric functions of several variables as an application of the theory. Since ordinary theory of integrals of single-valued functions is formalized under the name of the de Rham theory, by modifying this theory, we will constuct a theory suitable for our purpose in this chapter. As the key to the de Rham theory is Stokes theorem, we will start by posing the question how to formulate Stokes theorem for integrals of multi-valued functions.