A very interesting integral with aesthetically pleasing solution development
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- čas přidán 21. 08. 2024
- Today's integral looks absolutely gorgeous and yeilds a very cool final result in terms of our favorite transcendentals.
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I love when he said ”it’s integral time” and integraled all over the integral. Truly one of the integral moments of 2024.
You're the only one who didn't get a heart
@@edcify8241you just had to jinx it.
I thought we were done with this
damn this video was so beautiful that i lost my edging streak.I love math.
Yup CZcams is definitely recommending my videos to the right audience.
bro
guys i think he likes likes math
Hi,
"ok, cool" : 1:06 , 5:27 , 6:37 , 7:37 , 8:12 ,
"terribly sorry about that" : 2:20 .
It feels like it'd be a ton easier writing cos(x) = Re(exp(ix)), packing the exponentials, completing the square and then invoking some mild holomorphic property to make the imaginary shift in the integration variable. After that it's just gaussian integrating.
You could even do that at the step before, changing the sin into a complex exp.
Still, watching Feynman's trick at work is always nice, keep it up.
Or you could just use the complex definition of cosx and then complete the square in both Integrals and create two erf functions which solves the integral in like 2-3 lines
can you explain the holomorphic shift? what i get when completing the square is Re(exp(-(x^2-i/2)), how do you turn this back into a normal guassian without invoking complex integrals
@@aryaghahremani9304you should get something like exp(-(x-k*i)^2) inside the integral, for some k I can't tell right now.
Then you'd want to do the change of variables u = x - k*i, to get back to the normal gaussian integral, but to do so you must note that the 2 "line pieces at real infinity" integrate to 0 and that the integrand is holomorphic.
You can also just do the change, replace the limits and hope it works, but to justify it you need this very mild complex analysis, which is what I was referring to.
@@lakshay3745 I agree: I felt as if the use of Fenyman's trick was strained. My first reaction was to express the cosine in complex exponentials, as you suggest. But maybe because I never learned it in school so I find it non-intuitive.
And at that point it's just the Fourier transform of the Gaussian evaluated at 1
WOW, what a cute ODE, i loved how you used the feyman trick there, i couldn't wait for you to mention the gaussian integral for that -u² on the e ahahahaha
you should try the book of almost impossible integrals. Its a joy to solve those!
I liked how the e^(-u^2) term kept absorbing the u's
Hi Kamal
I've been watching your amazing 'integral' videos, and the more I watch, the more I love them, and I see you using that amazing Feynman's trick that's really cool.
Indeed you make my 'integral moment' today at sunrise here in India
ODE is smart, i love how feynman's technique is becoming an actual method of solving integrals rather than just a cool trick these days
i solved the cosx e^(-x^2) integral by letting cosx=Re(e^ix) though which is pretty cool as well i guess
Free okay cool buttons:
1:05
5:27
6:36
7:37
8:11
Counting how many times bro said "cool".
Integration by parts
Change of variable u=sqrt(-ln(x))
Series expansion for cosine
Change order of integration and summation
Change of variable v = u^2 , to get Γ function
(Γ function can be also expanded)
Very impressive integral. Thanks for featured solution.
Both the solution and the technique are beautiful.
@ 4:49 for this part I would have used the complex expression for cos(u) and then used contour integration to get the rest of the answer
people say im weird cause to me this is fun, i think their werid for not understanding that this was fun
Thank you!
First time i accually manage to solve one of your monster integrals, lets go
I'm proud of ya bro
one of the few times i could've solved it by myself
I liked how we never needed to go back to the original
Great thank you Sir
Very cool.
Ok cool!
Beautiful one!
Io ho fatto così...t=√(-lnx)..risulta e^(-t^2)cost integrata da 0--->inf...poi feyman I(a)=..cosat...risulta,in sintesi I=√π/2e^(-1/4)
Why not complement it with a graph? In most math calculations, a graph speaks a thousand words and can be very intuitive. Thank you all the same.
When you get in the next step, please put some explanation between steps. Because we need to get the clearance solution. we are not all proficient .thank you
So you weren't listening to the audio???
@maths_505 Absolutely, because mathematics is a language in itself, so l asked you, we need more steps before you get in another step,
@@user-zg8ny5tp4g to waste more time??
@maths_505 Why do you think that 20 minutes is not enough ??
@user-zg8ny5tp4g I just think that rambling on about basic algebra is gonna be extremely boring. The level of math here is something that my target audience is sufficiently familiar with.
Beautiful aesthetically pleasing result, use of differentiation under the integral sign which led to a lovely first order ODE 😍
Exactly what I needed on my Friday morning.
Could you find and do a beasty integral that evaluates to Digamma(G/sqrt(phi))?
Wow that is some request! I'll try my best.
@@maths_505 I know that's a very tough request. 😂 I don't expect you to get very far with it; as far as I'm aware crafting a (non-trivial) integral to yield a specific result is some near-impossible guesswork. I suppose that's why it's extra-special when beautiful results DO come out 😊
I have read about norms the other day
(barely understood anything)
But I got wondering
how to find a general formula for the integral from 0 to 1 of
The nth root of (1-x^n)
Substitute `x^n = u` and notice that you get a kind of beta-function
Hi bro, can you do the intégrale from 0 to 1 for "ln(1-ln x)"
Aight
hi please why did you plug in ln c rather than c
Ooooook coooool!!!
Nice😊
i thought you sove it by complex nos
Keepin it real this time
Без Фейнмана не обошлось))
Ah, but is the final result irrational?
mane wtf😭😭