That is, And so, By (1), See also The Artistry of Mastery. Exercise-1 Completing the ..read more

Introducing Feynman’s Integral Method Michael Xue Vroom Laboratory http://www.vroomlab.com Indiana Section MAA Spring 2024 MEETING Marian University – Indianapolis April 5-6, 2024 “The first principle is that you must not fool yourself and you are the easiest person to fool.” — Richard Feynman Feynman’s Integral Method Leibniz : “” Feynman: “To evaluate hard differentiate under the integral sign!” Two algorithms implementing Feynman’s Integral method Algorithm #1 Reducing Algorithm #2 Generating from a known definite integral Example-1 Evaluate Example-2 Evaluate Example-3 Evalua ..read more

Feynman’s trick, also known as parameter differentiation under the integral sign, is a powerful mathematical technique used for simplifying complex integrals. By introducing an auxiliary parameter into the integral and then differentiating with respect to that parameter, Feynman’s method often transforms difficult integrals into more manageable forms. It is a versatile tool in physics and mathematics, making the evaluation of certain integrals straightforward where traditional methods might falter. Introducing Feynman’s Integral Method Michael Xue Vroom Laboratory http://www.vroomlab.com Indi ..read more

In Deriving the Extraordinary Euler Sum , we derived one of Euler’s most celebrated results: Now, we aim to provide a rigorous proof of this statement. First, we expressed the partial sum of the left-hand side as follows: This splits the partial sum into two parts: one involving the squares of even numbers and the other involving the squares of odd numbers. Simplifying the even part gives us: Rearranging terms on one side, we obtain: Since converges to (see My Shot at Harmonic Series) converges to to prove (1), it suffices to demonstrate that or equivalently, Let and we have ..read more

For Let I see and Since for I have so ..read more

Prove: For any positive number , let Since is a monotonic increasing function: we have That is, or By the definition of : we also have Combining (1) and (2) gives This implies that In other words, See also The Sandwich Theorem for Functions 2 ..read more

Evaluate This integral is known as the Dirichlet Integral, named in honor of the esteemed German mathematician Peter Dirichlet. Due to the absence of an elementary antiderivative for the integrand, its evaluation by applying the Newton-Leibniz rule renders an impasse. However, the Feynman’s integral technique offers a solution. The even nature of the function implies that Let’s consider and define We can differentiate with respect to Hence, we find Integrating with respect to from to gives Since and , we arrive at It follows that by (*): Show that From the inequality and ..read more

For given functions and The given condition gives and It means Since we have That is, Or, And so, See also Sandwich Theorems and Their Proofs ..read more

Show that This integral is renowned in mathematics as the Gaussian integral. Its evaluation poses a challenge due to the absence of an elementary antiderivative expressed in standard functions. Conventionally, one method involves “squaring the integral” and subsequently interpreting the resulting double integral in polar coordinates. However, an alternative approach, which we present here, employs Feynman’s integral technique. The even nature of the function implies that Let’s consider and define We can differentiate with respect to Given a differentiable function on with derivat ..read more

Given is continuous on and satisfies the following conditions: [1] [2] [3] has continuous derivative on Prove: The given premises is a continuous function on ensures the existence of the definite integral and the antiderivative of Denoting the antiderivative of as we obtain We also deduce from [3] that is continuous on Combining [3], [1] and (1), is continuous on Additionally, as per [3], is continuous on Together, (4) and (5) give is continuous on Consequently, exists as well. Now let’s examine , where . We have is the antiderivative of This theorem serves as the ..read more