It feels like it has been a long time since I last wrote an article. Now that the college entrance exams are over, I am continuing my research. First, let me summarize some results from before the exams.
This article is about something called “differentiation under the integral sign,” which is a very interesting and useful method for evaluating definite integrals. Here, I have arbitrarily named it “Feynman’s Integration Method” because I first encountered this technique in Feynman’s autobiography, Surely You’re Joking, Mr. Feynman!. Of course, Feynman was not the inventor of this method; he simply loved it, was proficient in it, and recorded it in his autobiography. What exactly is the story? I won’t say too much yet; please read the following excerpts from Surely You’re Joking, Mr. Feynman!.
Surely You’re Joking, Mr. Feynman!.txt
Scenario 1: Arguing with Mathematicians
That book also 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. So because I was self-taught using that book, I had peculiar methods for doing integrals.
The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. 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.
The book mentioned is Advanced Calculus by Woods, which was given to Feynman by his high school physics teacher because Feynman was always causing trouble in class. From this description, we can see that this method is like a “secret technique” that often succeeds by taking the opponent by surprise! Seeing this, I was deeply attracted to the method and determined to learn it.
Scenario 2: The Atomic Bomb Story
But my luck was often good, and when they explained their difficulties to me, I would blurt out, “Why don’t you try differentiating under the integral sign?” In half an hour, the problem they had been struggling with for three months was solved. Thus, using my unique mathematical tools, I also made a small contribution. After returning from Chicago, I reported to everyone: how much energy was released in the experiment, what the atomic bomb would look like, and so on.
See, Feynman is always like Peter Pan, making people marvel at his genius! I think this is inseparable from his ability to learn various methods and then condense them into the essence of thought for application.
Scenario 3: Accepting a Challenge
One time I boasted, “I can do by other methods any integral that anyone else can do by contour integration.”
So Paul Olum challenged me with a magnificent, hellish integral. He started with a complex function he knew the answer to, took off the real part, and left the imaginary part—it resulted in a problem that was impossible to do except by contour integration! He always deflated me; he was a very smart guy.
Here, “contour integration” refers to using complex analysis to evaluate real integrals, which is also an excellent method, but Feynman said he “never did learn” contour integration. From this, we can see that “Feynman’s Integration Method” is not omnipotent! However, this gave me even more of an impulse and determination to accept the challenge!
I like technical tricks because they are truly interesting, but I don’t just play with them as tricks; I study them, find ways to broaden them, and try to turn them into a research philosophy and methodology. “Feynman’s Integration Method” is exactly such an interesting thing. To find the definite integral of a function, we first differentiate it with respect to a parameter, and then integrate twice with respect to different variables. In this way, the direct integration process is transformed into three steps: “differentiation — integration — integration.” It might seem like making something simple more complex, but it is precisely through this operation that unexpected results are achieved! In my view, this is a tactic of “abandoning first to take later” and “playing hard to get,” which naturally leads to brilliant results!
Enough with the words, let’s look at a specific example:
Evaluate the integral: F(a)=\int_0^1 \frac{x^a-1}{\ln x}dx
Differentiate with respect to a: \frac{d F(a)}{da} = \int_0^1 \frac{\partial}{\partial a} \left( \frac{x^a-1}{\ln x} \right) dx
Since the latter part is only differentiating with respect to a, we treat x as a constant and a as the independent variable. Because \frac{\partial x^a}{\partial a} = x^a \ln x, we obtain (Integration):
\frac{d F(a)}{da} = \int_0^1 x^a dx = \left[ \frac{1}{1+a} x^{a+1} \right]_0^1 = \frac{1}{1+a}
Therefore (Integrate again): F(a) = \int \frac{1}{1+a}da = \ln(a+1) + C
There is a constant C here, which needs to be determined through a special case. When a=0, the original integral becomes: F(0) = \int_0^1 \frac{1-1}{\ln x}dx = \int_0^1 0dx = 0
That is, 0 = \ln(0+1) + C, so C=0. In other words: \int_0^1 \frac{x^a-1}{\ln x}dx = \ln(a+1)
This example basically contains all the procedures of “Feynman’s Integration Method.” It can be seen that by winding around like this, we actually arrived at the correct answer quite easily. This is the benefit of “simplifying by complicating”! Dear readers, if you haven’t fully grasped the process yet, please read this example carefully again; I believe you will gain something! If you still feel like you haven’t had enough, then in the next article, I will discuss the details of “Feynman’s Integration Method” with you more thoroughly.
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