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- เผยแพร่เมื่อ 14 ต.ค. 2024
- Yes yes I too am sad about not using contour integration this time but for the sake of a teaching exercise, I believe it's necessary to wander among possible solutions as efficiency can be highly appreciated.
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That moment you go so insane solving hard integrals that you end up quoting Mourinho 12:38
Calculus
Heritage
You can take the derivative of the beta function with respect to u, - the derivative of the beta function with respect to v, and then use gamma to conclude, but your approach is most likely much more efficient.
If we are writing about efficient approaches he could use integration by parts at 3:43
and would be even more efficiend
An even more efficient method would be to use Wolfram Alpha
@@stefanalecu9532 Not necessarily
Try to calculate integral of
x/sqrt(e^{x}+(x+2)^2)
and for d^n/dt^n (1/sqrt(1-2xt+t^2))
Wolfram Alpha result is far away from being efficient
Mr. Kamal is really Mathematician of Integration, because he is thinking about his next two or three steps so fast. Thanks for this solution.
There are many mathematicians man Kamal is just one of them😊😊
@@UnknownGhost97 give me some top names Subramanya
I happen to love complex analysis. The more the better. 😂
There are more hidden treasures in this integral. For instance when we generalise the integrand to sqrt(x^a/(1-x)), we get for a = odd OEIS A002549(a)/A002550(a) *Pi and for a = even we obtain sqrt(Pi)/Gamma(n/2+3/2) * ( ln(4)*Gamma(n/2+1)- i^n*Stirling1(n/2+1,2) ) :-)
Absolutely delightful
I am so grateful for this channel existing because you're fun to watch, you got that level 100 gyatt skibidi rizz
I'm old so I don't understand what you said but I know it's a compliment so I am eternally grateful.
If an integral is covered by kamal it is NOT simple 😭
in his terms it is simple😅
It would be great if you do more videos about interesting problems in real/complex analysis other than integrals. There are some very interesting infinite series, infinite products, and limits I've seen on MSE that could be worth making a video on. Some more variety would be nice. Just a suggestion. I love your content bro!
@@violintegral just as I was evaluating an infinite product as sleep continues to evade me 😂
@@violintegral also, nice hearing from you after such a long while.
Int zero to one of 1/(x-x²)½ can equally be evaluated using the beta fonction. But I like too the trig substitution.
Had to pause to figure out just exactly how the difference of the squareroot of inverse fractions could be simplified in such a way, that’s really neat
Hi, I always enjoy wathing your videos!
I have come across with a beautiful identity :
Σψ(n)/n²=2ζ(3)-ζ(2)γ, where ψ(z) is digamma function, ζ(s) Riemann zeta function, and γ Euler-Mascheroni constant.
And I think there must be a more general form, So would you treat the kind of the sum of digamma function? It will be exciting.
Thank you!
At 2:36 that's just (d/ds)(B(s,2-s)) with s=3/2, the rest is way easier.
😲 wow didn't noticed it
Well, that simplified nicely. I used the beta function myself to solve it, but this is very satisfying.
As always, very satisfying development and an awesome video. But was there calm music playing in the background while you were recording the voice for the video?
Nah that was just my ac😂
substitute 1/(1-x) = t, define f(k) = Int. (t^k)/(1+t)^2 from 0 to infinity.
1/(1+t)^2 = D(-1/(1+t))=-Sum (-1)^k* k* t^(k-1), k from 0 to inf. So t/(1+t)^2 = -Sum (-1)^k * k * t^k = Sum (-1)^k / k! * gamma(k+1) * (-1) * k * t^k.
f(k) = Int (t^(k-1)) * t/(1+t)^2, use ramanujans master theorem to get f(k)= pi * k / sin(pi * k), calculate f'(k = 1/2) = pi
Substitution y=sqrt(x/(1-x))
Integration by parts with
u = yln(y) dv = y/(1+y^2)^2 dy
Way ahead of ya
I think Integral 0 to infinity of ln(u)/(1 + u^2) only converges as a Cauchy PV of some sort. Limit s-> 0+ of integral from x to 1/s of same.
A=dB(1+s,1-s)/ds
| s=1/2
(G represents for gamma function)
dG(1+s)G(1-s)/G(2)/ds
=d[sG(s)G(1-s)]/ds
=d(πs/sinπs)/ds
=(πsinπs-π²scosπs)/sin²πs
| s=1/2
A=π
Very interesting. Thank you.
loved the approach
Thanks
Is this your first comment, I haven't seen you around here?
@@maths_505 yeah I'm not commenting much but I'm watching a lot of videos.helps me improve my mathematic vision
Hi,
11:31 : d \theta , 12:30 : fixed ,
"ok, cool" : 0:27 , 2:16 , 3:24 , 6:24 , 9:09 , 11:10 , 11:53 ,
"terribly sorry about that" : 1:49 , 2:06 , 2:52 , 4:30 , 4:34 , 7:38 , 10:48 , 12:32 .
Question, at 1:31, why does x+tx=t imply that x=t/1+t ?
You started apologizing for not apologizing properly, I didnt know you were canadian 😂
Unrelated question: is there a mathematical notation (or a commonly used function) to describe a sequence of X equal but non periodic digits?
For example if i want to ‘build’ a number with five 2, i just write 22222 but if i want to write one with one million of 2 (maybe followed by other digits) is there a way to do it?
You can use a summation for that, something like (latex) 2\sum_{n=1}^{P}{10^{n-1}} where P is the number of digits
2:03 how can we decide when to take the limit forl the left and from the right?
My guess would be that, in this instance, 1 from the right is not in your integration interval whereas 1 from the left is.
Just imagin you’re adding infinitly small rectangles, here the rectangles are for x from 0 to 1. If tou let x approches 1, while the rectangles are between x=0 and x=1, the rectangles are approching x=1 from the left, but if the rectangles approche 0, they approche it from the right
You had us in the 1st half. You had a part of me in the second half. Rest was nullfied ..... >.
"Terribly sorry about that" is what reminds me it's Kamal
Should I put X=sin²θ...then by parts
after3:38 ,you can use integration by parts.
What program do you use
Been watching for months now and I never really commented so here :)
x/(1-x)=t..lnt=u...risulta I=4(1-1/9-1/25+1/49+1/81-1/121-1/169...ma forse ho sbagliato dei segni,perché il risultato è π...ricontrollero,boh .ho rifatto i calcoli e risulta I(sechx) -inf +inf=[2arctg(th(x/2))]=2(π/4-(-π/4))=π
ookay cool!!
WOOHOO WE'RE IN A CULT 🙌🎉
Magnificent
we're not gonna make it with contour integration with this one
O..k.. cooool!
4:09 that u looked a lot like a phi. it's always phi, isn't it?
it would be cool if one of your integrals had a solution of pi * e * gamma * ln(2) ;)
You could use the Fundamental theorem of Engineering to simplify that a bit more
π=3
e=3
sinx=x
cosx=1
e=2
i≈1
You could use the Fundamental theorem of Engineering to simplify that a bit more
π=3
e=3
sinx=x
cosx=1
e=2
i≈1
@@alphazero339 I see....you're a man of culture as well
@@alphazero339 Did you just approximate i to 1?
@@stefanalecu9532 i⁴ is 1 just as 1⁴ is dont overcomplicate things
Spoke like a lion
I hated integration by parts. I use whatever it takes to avoid it.
where's Mr France? I need my timestamps 😡
bro just called me mf :/
"simple"
boringg i want some complex analysis 😒
You will have contour integration within the next few videos!
disliked because of cursive pi by 2
Evil
Tau is better anyways...
At every single step, i couldn’t unsee how the beta function was able to help us. I was like « come on please you love this function i know you saw it you had like 10 different ways to just deus ex machina this monster in 45 secondes » but i knew you had a plan. And I wasn’t disappointed 🫡