A ridiculously awesome infinite power tower integral!!!

Follow me on Instagram:
maths.505?igshi...
to support the channel:
/ maths505
This power tower integral was EPIC and the perfect opportunity to introduce the Lambert W function.

[@maths_505]

Correction: that's eta(2) at the end of the video. Apologies for the error. Here's the write up for the proof of the reduction formula I used in the video. Don't forget to follow for more awesome write ups on the gram: instagram.com/p/CtpX1B8tIcc/?igshid=MzRlODBiNWFlZA==

[@oskarrask9413]

#hustle #neverstop :)

[@maths_505]

@@oskarrask9413 said it before and I'll say it again every f**king time😎😂

[@mathalysisworld]

Can you prove that integral of lambert W function from 0 to e is e-1 using its infinite series expansion?

[@bogdancorobean9270]

I'm so glad people still start random math channels and find ways to provide cool content like this.

[@manstuckinabox3679]

"We're going to invoke the Lambert W function". You're invoking my Vietnam flashbacks of all the implicit solutions to differential equations I could've solved with this function. 6:17 I bet it involves Infinite Series. 6:26 I'M DYING LOL

[@maths_505]

Bro this comment made my day😂

[@jonathan3372]

Ah yes, taking the series representation of Lambert W function out of the pocket like it's just another day... (got to confess I never knew there was one in the first place)

[@maths_505]

I've got deep pockets when it comes to series expansions 😎😂

[@pureatheistic]

Well, what do you carry around in your math pocket on a regular day?

[@maths_505]

Series expansions, some nice contours to waltz across, the Leibniz rule in case I get bored, the gamma function, its cousin the beta function, its older brother the digamma function and in case the going gets tough, the whole family of polygamma functions and ofcourse some more special functions for special occasions.

[@Aditya_196]

😂😂 exactly like bro introduces it so casually I feel like it's normal for all the viewers to already know about it

[@ves7y775]

One word : wow

[@robertsandy3794]

Awesome result. You make it look so easy. Lambert W is one of BPRP's favourite functions

[@alielhajj7769]

I kept saying wow every 10 seconds

[@western977]

Absolutely mind-blowing! 🤯 I just recently discovered your channel. Now I enjoy every video. You are awesome! 😉

[@agrajyadav2951]

Holy hell that's so cool, how the hell does pi sneak into these things!?

[@nicogehren6566]

very well done

[@MrWael1970]

Wonderful, it is a magic 😊. Thank you

[@ProfessorWumbo]

Wow. This was beautiful.

[@gmnotyet]

Infinite power, indeed.

[@btd6vids]

That was insane

[@romanvolotov]

so cool!

[@filipsperl]

"Lambert is dead" 😂

[@Ash.bourne]

Iam really onto being tricky as u in math , u got any advice or a source i can have the good explanations to what i seek in each genre of math ( like : series , integration , differentiation, and so on .... ) .

[@ilmaio]

Ouch. It hurted.

[@eduardoeller183]

Great video!!!

[@maths_505]

Thanks

[@nickruffmath]

Awesome

[@sergten]

Wow!!!

[@valdodevirus]

Sorry for a dumb question, but IMHO (x^x) ^ (x^x) ^ ... equals to 1 when 0 < x

[@lucidreconalt3229]

Yk what Kamaal I'm done this this I've been watching for months and I still cannot solve this differential equation. Can you PLEASE write out the letter u next to the letter n and show me how to differentiate between the 2 of them so that I can finally maybe integrate these techniques into my daily life because right now I CANNOT understand what in the hell you mean when u write lu(x) Anyway much love ❤ :)

[@maths_505]

Oh that's how we write the natural logarithm on my planet. Sorry I'm still getting accustomed to your earth writings😂

[@hrishikeshaggrawal]

soo... could you prove the sum in the bazel problem is exactly twice of this power tower?

[@grantofat6438]

Doesn't the result depend on the value of x?

[@CryToTheAngelz]

Shouldn't that be eta(2) at the end?

[@maths_505]

Indeed it is. My apologies

[@aliaujla110]

Suppose we remove upper & lower limits.

[@mikeoffthebox]

Lambert is dead - long live Lambert!

[@zunaidparker]

Mind boggling that this even has a solution. Does this technique work for the clarification power tower x^x...?

[@maths_505]

Yeah it should. Only difference I see is getting W(-lnx) as the integrand. The series expansion and integration is then straight forward.

[@zunaidparker]

*classic power tower. Damn swipe typing!

[@maths_505]

​@@zunaidparker it's cool. Better than writing eta(1) right😂. Apologies for that....its been a pretty exhausting day

[@oo_rf_oo8824]

@@maths_505 isn't that .... exceeds the redius of convergence of the series expansion?

[@ekxo1126]

I think you have to pay a lot of attention to the convergence of the taylor series of the W function there

[@Maths_3.1415]

Where are you from? Aap kis desh se ho?

[@maths_505]

Apkey sath waley desh se bro. (This response was also generated by a mysterious AI)

[@Maths_3.1415]

​@@maths_505 I cannot believe you speak hindi (my mother tongue)

[@Maths_3.1415]

I think you are from America because your accent is just like American's accent

[@maths_505]

@@Maths_3.1415 nah bro I'm Pakistani and in Lahore, Pakistan.

[@Maths_3.1415]

​@@maths_505but your English is damn nice 😂

[@padraiggluck2980]

Why isn’t ln (x^x)^y = ln (x^xy) = xy ln(x)?

[@maths_505]

It is but the objective was to get an explicit form for y in terms of x so I only took the y down initially. However you can take the x down too and get the same result.

[@karma_kun9833]

300 like

[@pureatheistic]

You proved your great with series, but i didnt see any proof of the first claim of being good with the ladies. Only that your some amount less good with the ladies as you are with series by your own word. I 0refer having as few assumptions in my proofs as possible, so....where the receipts?

[@friedrichhayek4862]

You should stop writting the n as u

[@PotatoImaginator]

Very cool