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A New Perspective on Calculus: Feynman's Integration


Over 300 years have passed since integration became a part of us. People in the present are carrying on what Newton and Leibniz began. From trying to find the digits of π to the nearest accuracy to Zeno's Paradoxes to now the relating heat equation to the partial differentiation, calculus has been with us in one way or another.

Richard Feynman, another important scientist like Newton, is who I want to focus on. He is well-known for a wide range of contributions, including Path Formulation (for quantum mechanics), Parton Model, Quantum Cellular Automaton, etc. He spent the majority of his career as a theoretical physicist.


Before I talk about what he did that made calculus a tad bit easier, we need to talk about another interesting topic, The Gaussian Integral, and how it is related.



GAUSSIAN INTEGRAL


This function is the Gaussian Integral or Euler-Poisson Integral. This might look like any other function but it comes with more intricacies. Functions like these CANNOT be integrated. It is referred to as the error function. Of course, now it is possible. Everything is explicable, to a certain extent.


RELATION BETWEEN GAUSSIAN INTEGRAL AND FEYNMAN'S INTEGRATION


An integral like the one shown shown above now has an answer with the help of polar coordinates, cartesian coordinates, Laplace's method, etc. but most importantly, it can be solved easily using Feynman's integration.


FEYNMAN'S INTEGRATION

It refers to the method for resolving specific kinds of integrals, especially those that arise in the context of theoretical physics and quantum mechanics.


“I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.” -Richard Feynman


The integration method developed by Feynman involves deftly utilizing the concepts of differentiation and integration to reduce the complexity of integrals. Instead of taking a methodical step-by-step approach, Feynman urged thinking imaginatively and visually about the behavior of the integral and coming up with ways to represent it graphically.

It is crucial to remember that Feynman's strategy is not a formal mathematical technique but rather an intuitive technique used by physicists to resolve difficult integrals that frequently occur in quantum field theory and related fields of physics. Even though it might not work for all integral types, it has shown to be useful in situations where more conventional approaches might not be as efficient.


HOW FEYNMAN'S INTEGRATION WORKS


The idea of differentiation under the integral sign, which allows for the simplification of some integrals, is one of the fundamental ideas in Feynman integration. When dealing with integrals that depend on a parameter, this method is particularly helpful. Sometimes the total integral can be made simpler by differentiating with respect to the parameter and then integrating.


EXAMPLE :




HOW CAN THE GAUSSIAN INTEGRAL BE SOLVED USING FEYNMAN'S TECHNIQUE?


It can be solved multiple ways using Feynman's technique.


EXAMPLE 1:


EXAMPLE 2:



There are multiple ways now to solve the error function, and Feynman surely made some of them easier to solve.

“I think for lesson number one, to learn a mystic formula for answering questions is very bad.”


In essence, Feynman's integration technique serves as a reminder that innovation frequently results from unconventional thinking and that the search for elegant solutions to challenging problems is a hallmark of scientific advancement.


-Anushka S. World of Women in STEM



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