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Feynman Integral Method - Explained with a Simple Example

Popularity:934 ℃/2024-08-01 13:08:54

I brushed up on a video again and again today, and I wanted to go to bed (I stayed up late last night), but I couldn't sleep again near lunch, so I had to water down an essay to pass the time.

What is the Feynman Integral Method?

Let's look at the official explanation first:

The Feynman integral is a computational method for solving definite integrals of functions of a complex variable, proposed by Richard P. Feynman. The method is particularly well suited for dealing with path integral problems in physics, but it can also be applied to the calculation of integrals in other fields.
In the Feynman integration method, the product function is usually expressed in the form of a path integral, where the integrating variable is usually time or some other continuous parameter. The core idea of the Feynman integration method is to convert complex path integrals into simple line integrals, simplifying the computational process through the use of complex function theory and fractal geometry.
Specifically, the basic steps of the Feynman integral method are as follows:

  1. Converting path integrals to integrals of functions of a complex variable.
  2. Chunking of integrals using fractal geometry techniques.
  3. Perform a simple line integral for each piece.
  4. Finally, all the line integrals are summed to obtain the value of the original path integral.

A significant advantage of the Feynman integration method is its computational efficiency, especially for path integral problems that are difficult to deal with directly by conventional methods. However, it has some limitations, e.g., it requires certain analytic properties of the product function, and for some types of path integrals it may be necessary to use numerical rather than analytic methods.


At a glance, it's unreadable. Since I'm not a math major yet I haven't taken complex functions and I'm not sure if I want to take them in the future. However, I can understand the process of this video that I swiped today. I will summarize my understanding with an example below:

Let's look at the question first.


How should this integral be solved? Most people would probably use the integration by parts method, but the genius Feynman came up with an even more unusual solution - Feynman's Integral Method

the process of solving a problem

First, we can multiply the producted function by a\(e^{-ax}\)At this point, let's look at this new equation

At this point we just need to find this new integral and then make a = 0 to find the original integral.
What about this integral? We can take a derivative with respect to the parameter a. According to the differential theorem of Lebègue's integral: if the function f(x, a) is differentiable with respect to a and its derivative\(\frac{\partial f}{\partial a}\)converge absolutely on the interval of integration, then the integrals and derivatives can be exchanged in order. We can get:

Then, let's review Euler's formula:

Let's start by multiplying the left and right sides of the equation simultaneously by\(e^{-ax}\)Post-credit points are available:

At this point, we are pleasantly surprised to find that the imaginary part of the right-hand side of the equation removing i is not what we asked for.

Next, let's solve for the left side of the equation:

Note the final simplification, which employs multiplication by the conjugate to remove the imaginary part of the denominator
And we need the imaginary part of the integral:

It's available:

It is now sufficient to integrate this function with respect to a. We know that for the right-hand side of the equation, it has a corresponding formula for the integral, so we can obtain:

When a->infinity, the left side equals 0 and the right side equals\(C - \frac{\pi}{2}\); the solution is\(C = \frac{\pi}{2}\)Get:

Then bring in a = 0 and solve:

summarize

Ergo, it seems pretty simple when you look at it that way, but this solution is not something that the average person can come up with. In a nutshell, the idea is to transform an integral problem into another integral problem related to Euler's formula, and use the characteristics of Euler's formula to simplify the complexity of the problem. Well, then Feynman integral is here~!