One of THE craziest & most beautiful integrals in existence

WOW!
WHAT A WILD INTEGRATION!
Solution development using a bunch of my favourite tools and the integrals involved along with the final solution were indeed epic.
Reflection formula for the gamma function (part of solution development of another cool integral):
• My take on @MichaelPen...
The gamma-zeta integral:
• A quantum integral con...

[@maths_505]

Note: at the 6:50 mark it would've been better if I applied l'hopital's rule for the limit x->0. One differentiation of the numerator and denominator shows that the limit is indeed zero.

[@JO06]

Was thinking about that lol

[@tueur2squall973]

I think we all thought about that. I came to the comment section to say just that 😂

[@maths_505]

@@tueur2squall973 so used to dealing with those kind of structures I feel like just crossing them out to zero without any explanation at all😂

[@NittyLittyNiturzion]

I've been watching this channel for a bit now (since about 22k subscribers), and the problems have gotten more and more interesting, and has really made me cracked at integration

[@maths_505]

And it's only gonna get better mate😎

[@fhffhff]

π²$(0;∞)(1-ix²)²/sin²(πix²)dx=π(-x³/2-ix+0,5x-¹)(e^(πx²)+e^(-πx²))/(e^(π x²)-e^(-πx²))+$0,5πix-²ctg(πix²)dx-π (x+2$dx/(e^(2πx²)-1)dx)-0,75xln|co s(πix²)|+$0,75ln|(e^(2πx²)+1)/(e^(2 πx²)-1)|dx(0;∞)=-∞+∞i

[@BridgeBum]

Dr. Penn has a different focus, he's generally more interested in proofs than solving complex integrals. I enjoy both channels for different reasons.

[@mihaipuiu6231]

Nice demonstration! GOOD FOR YOU TEACHER 505!

[@MrWael1970]

Awesome solution. It is very interesting integral. Thank you.

[@vibaked]

I feel like you don’t get the opportunity to write the numeral 8 in your videos much, so I’m excited to see it make an appearance here

[@maths_505]

I barely get the chance to use any numerals in my videos so I in fact get nervous whenever I have to....
"Okay Kamaal.....its just the number 8....you just need 2 circles one slightly bigger than the other....gotta draw em tangent to each other.....but what if they aren't tangent?????...oh wait yeah no big deal.....but what if they're so off tangent that it looks weird!!!"

[@jkid1134]

​@@maths_505I write something like an S and connect the ends, highly recommend it

[@user-os6zy8ik3y]

I enjoyed watching your video. Utterly wonderful. I’m just curious about where you learned those divergent mathematical concepts! Thank you

[@maths_505]

I'm self taught so I just search up the internet for pdfs and videos. Alot of these integrals are homemade including this one.

[@balasubramanianravikumar5233]

Presently reading the book In pursuit of Zeta-3 by Paul Nahin. It is just watching a series of Maths 505 video clips.

[@EffySalcedo]

Nice one 👍

[@nicogehren6566]

Awesome!!!

[@onionbroisbestwaifu5067]

Ive always wanted to know what the squared norm of Gamma squared of (1+ix^2) on [0,infinity) was.

[@shanmugasundaram9688]

Integration by parts method works well in solving this integral.Very interesting.

[@maths_505]

Yeah. I thought about a series expansion but that gets clunky. IBP is quite elegant here.

[@insouciantFox]

Beautiful. Who needs real friends when you have Euler friends. They're much more complex and Γ be much more satistfying.

[@Amoeba_Podre]

What is the value of zeta(3/2)

[@johnporter7915]

This guy does this as casually as if I were explaining x value in x-1=5 to someone. 😂😂😂

[@maths_505]

And I also made a small casual mistake😂😂....check out the pinned comment.

[@johnporter7915]

@@maths_505 Haha. I meant this comment to be both funny and a compliment.

[@maths_505]

@@johnporter7915 yeah I know I just wanted to point out that limit evaluation 😂

[@maths_505]

@@johnporter7915 thanks mate

[@BridgeBum]

​@@maths_505Was that really a mistake though? You can show that the "exponential wins" using L'H if you want, you just used that lemma as a shortcut.

[@emanuellandeholm5657]

Cool!

[@TheDarkness__]

Hi! Is there anything where I can try to learn what the Gamma function is? Did you talk more deeply about it in a specific video?

[@maths_505]

Brilliant.org is a good place to start. Then you can find detailed notes and articles on the gamma function on the internet.

[@TheDarkness__]

@@maths_505🫶

[@ikarienator]

6:19 I don't get the argument to only differentiate x^3. I think we need to differentiate x^3/(exp(2pi x^2)-1)^2 there.

[@ikarienator]

Oh, I see what's going on. The f'/f^2 = -d(1/f).

[@sslelgamal5206]

I tried to solve it with Parseval-Plancherel's theorem for Fourier transform but unfortunately the Fourier of Gamma(ax^2) is not solvable! Same for Mellin and Laplace transform, also not usable, your method is the only one working 😅😅
It as surprising though, cause it kinda looks like Meijer or Fox's H function types!

[@maths_505]

The integral obtained after applying the reflection formula can be used to derive a melin transform for csch²(x²). All that's needed is integration by parts followed by calling on the gamma zeta integral.

[@maths_505]

But yeah no transform will work from the word go because of the gamma functions....that's the way I designed this integral😂

[@sslelgamal5206]

@@maths_505 Yeah, that's actually soo cool, I did not follow on the footsteps to arrive at the Csch step and wanted to go all out on Gamma's but this one works 👌👌👍👍

[@jonsmith8579]

wheres the gamma hook :(

[@manstuckinabox3679]

I think We can also use at 2:33 the inverse Mellin transform… you made a mistake and introduced me to this tranform now I refuse to shut up about it.

[@maths_505]

A mistake I will never regret my brother

[@giuseppemalaguti435]

In base ai miei calcoli... Doesnt converge... Boh???

[@maths_505]

Definitely converges bro😂

[@cycklist]

Clickbait title much?