Crazy Binomial Integral from Cambridge Integration Bee
2024 ж. 4 Нау.
29 685 Рет қаралды
Integral between - infinity and infinity of n choose x, from Cambridge Integration Bee 2023/2024 Round 2.
Integral between - infinity and infinity of n choose x, from Cambridge Integration Bee 2023/2024 Round 2.
When I saw this in the competition this integral made me physically sick
Wonderful video. I love how this result is analogous to the discrete case since holding n fixed and summing k from 0 to n is also 2^n.
Sophomore's dream: ∑ binom(n, x) = ∫ binom(n, x) = 2^n
😂
Cool! I love how it's equal to the discrere sum of binomial coefficients on naturals.
You really explained this integral super well and went into good amounts of detail for the *easier* steps, I really appreciate that someone not so familiar with all this stuff can come away feeling like they understood some of the ideas presented!
Great work! I like how at the end, you go through all the work to compute I_0 only to find out it's just 1
It's the cardinal sine sinc(x) regularized so that the zeros are at the integers but also regularized so that the integral is 1.
beautiful solution
Wooow what a beautiful solution 😮❤
so nice ❤
sweet and awesome (*smile)
That’s wild
i remember this one in the competition - i shouldve guessed the answer knowing the analogous discrete case!
This integral is only computable if you extend (n choose k).
Oh wow, that is astounding. Does this definition work for non-natural values of n? Meanwhile, it was very fascinating to see the Euler reflection formula emerging :)
It was mentioned that the identity used in the first step works for the Gamma function definition of the binomial as well, so I presume that everything works just fine for all real values of N as a result. Possibly even over complex N as long as the Gamma and binomial continue to behave properly.
@@HeavyMetalMouseWould that essentially mean everything else besides when the negative integer poles of the Gamma function comes up? That would be an extremely powerful result. I could understand that it would definitely hold over the x side of things but I wasn’t completely sure it would hold for non-natural n.
@@HeavyMetalMouse I’m in the process of testing the result in Desmos empirically, and while the convergence seems to work quite well for positive n, integer or otherwise, it seems to start diverging after n < -1.
Infinity
Cool! Can check using result by summing each row of Pascal’s triangle, as each row is just all possible values of n choose k for integer n, k
Is it really okay to completely disregard the fact that the Γ function is discontinuous at all negative integers and 0?
Of course. That set has measure zero after all.
@@frogkabobs Sure, but shouldn't it be checked that the improper integral converges in all those cases? I mean, the integral of 1/x dx between 0 and 5 doesn't converge and it only has one discontinuity.
Nice, nice, I have got a nice one for you. Show that \int_0^1 (x * [1/x]) dx = pi^2/12 , with [ ] being the floor function.
Hello 2^n
do you maybe have any resources for the gamma definition of a binomial?
If you know the factorial definition of the binomial coefficient then you can rewrite each of the factorials using the gamma function as defined at 5:25. If you’re still confused, I suggest you search “binomial coefficient wolfram mathworld” on google, and look at the wikipedia page for the gamma function 🙂
@@LukasTrak Thanks a lot!
What about integrating this between 0 and n ??
At 4:04 could you explain to me why you can assert this? Thanks. Also very good video, straight to the point.
Replacing the x+1 with x just results in a translation by 1 unit. As we are integrating between -inf and +inf, the total area under the curve would still remain the same, hence the integral becomes I_(n+1).
Can you use the gamma function for this as it is only defined for x>0?
The gamma function is defined for all x aside from non-positive integers, so yes
NO FUCNIKG WAY
Euler’s Γ is not a total function.
but you can expend it to a meromorphic function on the complexe field.
Not sure, if it was eligible, to change from factorial to gamma, to solve that shit integral being 1 in the end.
It is. That’s how you would extend binom(x,y) to x and y real in the first place.
first
Answer is 0. Next question?
?
@@frietvet Taking it literally, this function is only defined over the natural numbers
@@LookToWindward The function can be extended to all real numbers by using the gamma function in place of factorials. As we are integrating between -infinity and +infinity, we use the gamma function definition.
@@LukasTrak I know, it was a joke.
@@LookToWindward you found the most weird way to joke lol
OMG you could have shortened the video by 50% at least - many of the steps are trivial.
Nice! But can’t we just say that the binomial only makes sense when 0
Unfortunately, the binomial does ‘make sense’ for all real x using the gamma function definition of the binomial coefficient.
@@LukasTrakyou should say fortunately rather than unfortunately.