INSANE integral solved using 2 different methods (feat. Feynman's technique and Lobachevsky)
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- Опубліковано 7 бер 2023
- One of my favourite integrals so far and both solution developments were incredible.
Here's the result referenced in the video:
Using Feynman's approach: • Using Feynman's techni...
Using Laplace transforms:
• One of the most beauti...
Using contour integration:
• YOUR FAVOURITE CALCULU...
Bro's a menace with the dead jokes lately 💀💀
😂😂😂
"Today's solution development is based on an awesome result by some dead guy."
Jokes aside, I rank this integral as one of my top 5 favourites....it was gorgeous
@@maths_505 Oh yeah, I definitely agree with that! It was really beautiful indeed.
Personally, I liked the first one with the infinite sum better, as I find it quite logical and methodical rather than a plug 'n chug formula.
If there's something I hate most in Mathematics, it's formulas for integrals. They're just about as exciting as basic algebraic equations.
@@daddy_myers ahh......you're a man of culture as well😎👑
Though Lobachevsky's formula is alot more satisfying once you see the proof...I'll try to upload one in the coming days by generalizing the brute force application I've used here
Finally one using Lobachevsky😊
Few more coming up....including proofs
Gf wasn't too happy when I told her that this was the best thing I'd seen this day... (＠_＠;)
9:40 Here Laplace transform can be used if you prefer integration under integral sign
Mine was the first technique as well, can't wait for the proof to come out though.
Love this integral! What digital blackboard/whiteboard do you like using?
Just the default one on my phone
Looks like Samsung Notes.
9:40 This becomes extremely clean when you know the integrand is a product of two known Fourier transforms, one of a window function from -1/2 to 1/2, and the other of e^-|x|, then you just use Plancherel theorem instead.
Great 👍
Legend
Wow! To be honest this one goes kinda over my head 😂, but amazing solution
A thing that annoys me is that most Feynman's technique videos just casually swap infinite sums and derivatives with integrals showing no work that those operations make sense.
How to show that that sum equal cot(x) ?
For that you have to wait 2 videos😂
From what I remember, the proof of Lobachevsky’s integral formula involves something like the first solving method
Yup....that's how I proved it in a recent video too
How does the cotangent series converge?🤔
Wait for today's video
@@maths_505 Release time?
How about
∫ sin ( tan (x) ) / sin (x) dx
// limits from -∞ to ∞
?
😶😇😄
when ru gonna reveal your face ?! pleaseeeeeee!!
btw great integral .
It annoys me slightly that you don't justify that the integral is defined.
The same goes has you split happily the integral in two starting at 0. How does the function behaves in 0 ?
What happens at Pi/2 (mod pi) ?
We have a similar issue when you swap the sum and the integral. Are just these values finite ? Does it converge ?
Some integrals are pathologic enough that one is able to integrate them formally but are in fact undefined.
As such I prefer the second solution even if I am a bit suspicious about Lobachevsky's proof ;-).
Overall I trust intuitively trust that "everything is ok" and the result is accurate.
(I guess that one can use Dirichlet's integral as bounding function.)
Your concerns are indeed valid
We see that the limit of the integrand as x approaches zero is 1.....elementary calculus proves this.
As far as x approaches pi/2 is concerned, the sine function is bounded which further alleviates concerns....hence the switch up of limits using first Fubini's theorem and then the Leibniz rule.
And don't worry about Lobachevsky; I'll upload proofs in the coming days for both cases: 1) f(x+pi)=f(x) and f(x+pi)=-f(x)
Your pronunciation of “Lobachevsky” is quite fine. I would say, almost perfect
But yeah, he’s probably don’t give a fuck
Yes, agree.
Harder to find out was the meaning of modified 'Basel' and 'Strub' special functions from the holiday season video, which turned out to be Bessel (pronounced like vessel) and Struve, although fortunately their standard acronyms were used and Abramowitz/Stegun is your friend.
Pronounced: lo-ba-ch-ef-sky
The calculus tricks and narration are great, but... your proofs really lack checks of convergence of your improper integrals and infinite sums.. Same with when you permute sum and integral symbols.
You talk too fast.