Gamma function, Beta function, Digamma functions, Polygamma function, Zeta functions, oh well all of them are connected with Pascal Triangle, Fibonacci, Lucas, Bell, Bernoulli Numbers, Stirling Numbers and so on. Math has endless beauty. Indeed.
No matter whether n is an integer or not - as long as the stuff is well enough defined, so nothing like nCr(-1,x) - we can see ∫[x=-∞,∞] nCr(n,x) dx = Σ[x=-∞,∞] nCr(n,x) This is impressive. The only other function I can think of that shares this property is 0, but that's trivial. ∫[x=-∞,∞] 0 dx = Σ[x=-∞,∞] 0 dx I mean, things like x² and e^x would also work, but because they both diverge to ∞ for the sum over Z and the integral over R. It's when they're convergent that we get the real results of interest.
One of my close friends marked this integration bee, he was stunned and stumped when he saw this question. Its amazing how this is the continuous version of the discrete summation identity and holds as 2^n. In fact, one can quote this result given enough complex analysis tools and theorems (such as Wielandt's Theorem), using the AC of Γ(z) to all the complex numbers outside Re(z) > 1. The integral is f(n), which has to be continuous by analyticity. In the n discrete case, we get 2^n and so the whole integral has to be f(n) = 2^n by continuity.
Yes, he does make several assumptions so the videos *don't become too long* . e.g. he knows factorial is not a continuous function and quickly assumed using gamma function before deriving how gamma function ends up mapping to (n-1) factorial, which would be super helpful to younger audience of his channel. Even those who may have heard of gamma function may not know about its sophisticated properties such as the reflection formula, which was assumed and never derived in the video but critical for solving this problem. Lastly, he glossed over other issues with gamma functions within his integration range (e.g. poles at negative integers). Even after all these assumption and no derivations it's over 14 min video, so he's relying on 'exercise to the reader to derive these formulae/assumptions', to keep the videos short with just the most critical solution steps.
Hello proffesor i trying to solve this equation but without nothing the eq is xʸ=xy It looks very hard even AI and wolfram alph cannot solve it Can you? If you get the solution i wish If you make a vedio on it My greeting to you
