Calculus of a kind of improper integral

Abstract

The calculation of improper integral is of great importance. In this paper, we consider a kind of real improper integral using the method of complex analysis and prove the equation ∫ 0 ∞ x α ( 1 + x β ) x d x = π β sin α π β . Firstly we give the condition for the convergence of the improper integral ∫ 0 ∞ x α ( 1 + x β ) x d x when 0 < α < β , so we can see the integrand as a complex function. This function is defined on a simply connected set not containing zero to ensure that the function is holomorphic. We use three methods to prove the equation. In the first method, we choose a certain closed path as the boundary of the simply connected set, compute the integral along the path by the Residue Formula according to Cauchy’s Theorem, and obtain the value of the real improper integral. In the second method, we use Mellin transform, while the idea is similar to that of the Residue Formula. In the third method, we find that the path we choose becomes simpler by a variable substitution. The function that is integrated along the new path does not have the problem of multivaluedness, so we do not have to define it on a simply connect set. Moreover, using the method of complex analysis, we prove that the equation holds when α, β are complex numbers, and the condition is about the real parts of α and β, i.e., 0 < ℜ α < ℜ β . The real improper integral is just a particular case where α, β are real numbers. We compute the value of such kind of complex improper integral by some calculation and simplification, which is exactly π β sin α π β . We find the relationship between this integral and gamma function. The equation can prove a property of gamma function. We prove that the equation holds when α, β are complex numbers, and the condition becomes 0 < ℜ α < ℜ β . We hope that this can be used in the research of more properties of the gamma function. The proof of the equation reminds us of a way of calculating such kinds of real improper integrals.