I sat in on Physics X at Caltech for a year: This is where Feynman would take questions from freshmen and sophomores, and try to answer them. He didn't like the more 'formal' questions that upper-division and grad students would ask: he wanted questions driven by real curiosity.
I had a lot of fun in Physics X: One time I mentioned something that had bugged more for a long time: "Why is the gamma function generally considered the 'natural generalization' of n-factorial (n!)?". He didn't answer directly; instead, he turned to the classroom and issued a challenge: "Over the break, think about this: 'What is (1/2)!, the factorial of (1/2)?' OK, Merry Christmas!"
Three students actually thought about this over the break. One girl pointed out that (d/dx)^n[x^n] = n!; I thought something could be done with that. Another fellow expanded the factorial function in a complex power series about z = (1/2,0), imposing the conditions that (z +1)! = (z+1)*z! and also that 1! = 1 = 0!; he carried this out to more and more terms.
I picked up the idea about the repeated differentiation, and combined it with the fact that a repeated integral also incorporates the factorial function. (See
mathworld.wolfram.com/RepeatedIntegral.ht ml ; you'll need to remove the space).
For integer values, the relationship between n! and the nth repeated integral is just true. I co-defined non-integral values of n! AND non-integral repeated integrals so that the relationship would still be true. I then proved that relationship between the gamma function and the beta function (See:
http://en.wikipedia.org/wiki/Beta_function#Relationship_between_gamma_function_and_beta_function)
also held for my extension of the factorial. I was 95% of the way there, but I got stuck on one point: once I could figure out one value, for n = 1/2, I could immediately show that any continuous function that was defined in my way had to be the famous gamma function.
On the first Monday evening of the new calendar year, only 4 people showed up: the three factorialists and Feynman. Feynman was delighted at what we had found: It all reminded him of things he had worked on previously. I think he pointed out something very easy that I hadn't finished thinking through (basically integration by parts; I can't find my notebook to see why that would have been a problem), and then boom!, I had (1/2)! = sqrt(pi)/2; knew my definition of n! = gamma(n+1); and had a definition for non-integer iterations of integration and differentiation, all together.
Feynman dug it. Then he gave me a bit of advice: "Don't tell anyone. Just wait. Someday, someone will have a problem, and this will be the answer to it. Just bring it out then. And then when they ask you, 'How did you figure that out?', you just say, 'Oh, I discovered that when I was just a kid.'!"
I ran into Feynman at the Caltech cafeteria, and he invited me to join him for lunch three different times. This was the highlight of my year at Caltech.
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