The INCREDIBLE Malmsten integral 

this integral is zeroeth degree murder

that's a horror movie

man I know you do crazy integrals but this is something else bro this is a murder

I suffered the little death, in the Shakespearean sense.

so early there isn't even a thumbnail

I do believe it's time to face your destiny and tackle the Borwein integrals.

Finnaly legendery integral! Next Coxeter integrals! 🖤

I did not survive... Will try again tomorrow.

“Thank You Math 505” we all said in Unison

Is the integral related to the poisson kernel?

You could say that the quadratic was a known kernel

Coomer series coming in clutch! 🎉

I couldn't help but notice that alpha=0 limit exsit in that gorgeous answer, so I did it as a homework and got 1/2ln(e^-γ * pi/2) which is a nice bonus result for a special case of first integral at alpha=0

I survived, but now I carry with me the weight of the experience.

Hi, "ok, cool" : 2:23 , 6:24 , 14:54 , 20:55 , 22:28 , 22:57 , "terribly sorry about that" : 11:00 , 11:36 , 14:36 , 18:42 , 26:50 , 27:17 .

10:42 part is beautiful

Early in the morning and I'm watching such an integral instead of going to work!!! 😅

Breathtaking!

Great integral man

Shredder! Minor quibble. At ten minutes thirty, divide by 2i to get sine.

Me before watching this video: the polinomial in the denominator can be computed as (x+e^itheta)(x-e^itheta). now we define I(s) = int from 1 to infinity of log(x)^s/(x-a)(x-b). applying PFD. a = complement of b a = e^itheta. I(s) = int 1 to infinity of (1/b-a) * (log(x)^s/(x-b)) + (1/a-b) * (log(x)^s/(x-b)) . we split into I1(s) and I2(s). apply u = log(x) e^u du = dx e^u *u^s/(e^u-b) = u^s/(1-b*e^-u) so using some melliny mellin magic I1 evaluates to.. iuhun I'm just going to watch the vid now.

Hello! Is there a "simple" function whose integral has a very "complex" closed form? By a simple function, I mean a function similar to the one in the video, consisting of a few elementary functions, at most rational. By a very complex closed form, I mean expressions that involve various constants, raficals, logarithms, sums and multiplications etc. (ellipitc integrals, hylergeometric, gamma, zeta functions are not allowed, solution must be close). Thanks for any insights!

barely survived this one

The universe is now speaking to me

Will come back to this video when my math skills are better

Man, I remember when I watched Peyam solve this integral, you did it on 3x speed, this has been just beautiful, we should call you Magician Kamaal because you're pulling tricks left and right

I survived it! Dr Peyam did this integral on RU-vid a while ago.

Wicked awesome.

Can u cover Green's therom and scattering therom

This is far harder than the coxeter's integral. What makes it insane is that it's linked to geometric series, Fourier series, Laplace transform. Almost all the craziest analysis stuff where required to solve it. It was really a worthy opponent.

15:05 Im... still breathing...

awesome

Love it

I think I've survived

I like Kummer series. Exciting integral!

The same denominator of the integrand (x² +2xcosα + 1) appears in the result of the sum ∑ Rⁿcos(nθ), for n ≥ 0. (That is, if we let R=-x and α=θ) I wonder if this has any significance, of if it could be used to solve this integral in another way. edit: nevermind, after 4:00 you used a very similar argument

I really enjoyed this. But I have one question I dont understand why you wrote (2*pi)^(alpha/(2*pi)) shouldnt it be just (2*pi)^(alpha/2).

Also, a weirdly specific conclusion is that the integral of ln(lnx)/(x±1)^2 over (1,inf) diverges

Shoutout to one of the few world famous Swedish matematicians. We also have Ivar Fredholm and 50% of Harry Nyquist. The Norwegians of course have Marius Sophus Lie, who's on another level altogether. And the Danes have all the celebrated Physics/QM guys. The Finns have lots of great mathematicians too.

At 10:26-ish, you need to divide the upstairs and downstairs by 2j, not just 2, to get the complex sine definition. Ultimately you're still multiplying by 1, but the point stands 😉

My question: WHERE would you encounter such a monstrosity???

at 10:30 it should be divided by 2i, shouldn't it? ik it doesn't matter but still

Oh wow, didn’t realize this integral would invoke the Laplace transform and the Fourier series… wow

13:31 dont mind me, this is just my personal timestamp

Somehow I survived and honestly… I don’t know how

Now it’s time to prove that identity to ln(gamma)

I died, but I turned out fine as well

Where do you find all these integrals? Are they sent by the subscribers? I sent you two in the last few months.

You ruined my lazy Sunday. Will I survive? Greetings from Germany!

Yoooo I’m alive, but my god man

You must devide by 2i to get sin, not 2.

1) you forgot a - sign at 14:30 in the second sum 2) If you replace cos(alpha) with cos(alpha * x) in the denominator, is this related to Coxeter's integral?

Sorry Kamal but I died. However upon seeing the raw power of what killed me, Eru intervened and had me reborn

I'm okay

Again poor maskeroni canseling out😂

What am I missing: As alpha goes to zero, your answer approaches (1/2)log(2*pi). This is a positive number, but the integral should be negative. Huh?

I didn't know Yngwie had his own integral. (This is a guitarist joke.)

Wow.

Lordy. 🏆

anything

IDK if it was integral or the notation that killed me. :)

OK COOL

OK COOL❤

RIP my boy kamaal(I hope I am writing your name right)

i died

Unfortunately, I did not survive 😮‍💨

Er, I got the wrong channel

Nice and easy😂😂😂😂😂

I stopped following you at 9:00 , cause I couldn't follow.