Definite integrals containing a parameter

Solving Some Definite Integrals by Using Maple

This study uses the mathematical software Maple for the auxiliary tool to evaluate two types of definite integrals. We can obtain the closed forms of these two types of definite integrals by using differentiation with respect to a parameter and Leibniz differential rule. At the same time, we provide some definite integrals to do calculation practically. The research methods adopted in this stud...

Definite Integrals using Orthogonality and Integral Transforms

We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.

Feynman parameter integrals

We often deal with products of many propagator factors in loop integrals. The trick is to combine many propagators into a single fraction so that the four-momentum integration can be done easily. This is done commonly using so-called Feynman parameters. We rewrite the product of propagators 1 (A 1 + ii)(A 2 + ii) · · · (A n + ii) , (1) where A i has the form of p 2 − m 2. The sign of A i is not...

Multi-parameter singular integrals

A Lost Theorem: Definite Integrals in an Asymptotic Setting

We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant ...