Vardi Integral

[@weinihao3632]

Hello Dr.Peyam, the Kummer (Fourier) series (16:38) was derived by Prof. E.E.Kummer of Breslau and published in "Beitrag zur Theorie der Function . Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal), 1847(35), 1-4. doi:10.1515/crll.1847.35.1 " (page 4, first equation) sci-hub.se/10.1515/crll.1847.35.1 (C is the Euler-Mascheroni constant, l is the natural logarithm) Ihre Videos sind toll!

[@drpeyam]

That is amazing, thank you for the cool comment!

[@rot6015]

this is awesome, thanks for sharing

[@saulmendoza1652]

Nice publication.

[@pablojulianjimenezcano4362]

This is the most incredible integral I've ever seen!!

[@cycklist]

What an amazing ride that was! Thank you for introducing the Kummer series too, how interesting.

[@emanuelmartinez3585]

More series and integrals! OMG So much fun 🙊

[@Rundas69420]

I think this boi is on one level with 1/(tan(x)+(tan(x))^(1/3)). These integral-videos never fail to entertain and also remind me that I still have stuff to learn :D

[@robertreynolds1592]

The Hankel contour along with Cauchy's integral formula is very good at deriving the integrals in Prudnikov, Bierens de Haan, Gradshteyn and Grobner books.

Euler-mascaropne :) constant is a delicious one :)

[@karolakkolo123]

I am not sure but I think a general solution may exist (indefinite integral), and be written in terms of polylogarithm functions. Also, there is an intimate connection between the gamma function, zeta function, and the polylogarithms. The appearance of the gamma function in the answer gives more hope for the existence of the indefinite integral.

[@karolakkolo123]

I'm determined to find it when I wake up tomorrow

[@Anokosciant]

Did you find it?

[@karolakkolo123]

@@Anokosciant nope haha I completely forgot about this. I didn't even do integrals for a long time now. But I'll write this down somewhere and try it in some spare time

[@DevalMehtaAstrokidintraining]

We could eliminate the dependence on the gamma function by using the fact that Γ(z+1) = zΓ(z), right? In fact, doing that and then distributing the exponent, rather than using the properties of the logarithm to bring it outside yields (π/4)ln(π^2/8).

[@albertemc2stein290]

Great video Dr Peyam! But why are we allowed to use the power series at 4:35 with the constraint of |x| < 1 when at u = 0 -> exp(-2u) = 1?

[@drpeyam]

It’s an improper integral :)

[@albertemc2stein290]

@@drpeyam So the main integral is a limit problem and the divergent series just vanishes in the end?

[@neilgerace355]

1:20 Unleash the power of CHEN LU!

[@chandankar5032]

Wow ! Seems like you are after flammable maths

[@quantumcity6679]

Thanks.... Dr.peyam.....for this beautiful integral... 🤓....keep it up 😘...but can you please tell me when we can interchange integral and summation?...i mean the condition... ♻️🔊

[@juauke]

IINM, you can interchange 2 integrals or an integral and a sum (the latter being a special case of the former) if when you calculate the integral but with the absolute value of the integrand (the thing inside the integral without the dx), you get a finite answer. It's called Fubini's theorem (Fubini-Tonelli is probably more accurate but I'm not sure here) as indicated by πm. In mathematical terms, you'd get for the sum and integral case : let f be some measurable function. Let A be a σ-finite space. ∫_A means the integral over A. if ∫_A Σ |f(x)| dx < ∞ then ∫_A Σ f(x) dx = Σ ∫_A f(x) dx Voilà. Hoping this was clear enough :), if I did some plumbers please feel free to tell me, I'm always happy to learn more about Mathematics.

[@quantumcity6679]

@@juauke thanks....for this information ..... I like your presentation and math language that you had used here.... 😇..👍

[@juauke]

@@quantumcity6679 you're welcome :^)

[@quantumcity6679]

@@juauke appreciate... 😅

[@pedroalonso7606]

I think both gamma functions can be absorbed by a Euler Beta function, and given gamma(1/2)=sqrt(pi) it allows us to play with the other sqrt(pi) term.

[@GreenMeansGOF]

Equivalently, we have π*ln(Γ(3/4))-π*ln(π)/4

[@lucasdepetris5896]

Hello Dr. Peyam. I'm Lucas from Argentina and I have a question for you. Considering the arithmetical series A1, A2, A3...=1, 3, 5... Namely the odd numbers. Is there exist an Asub i term?? Is there a way to expand the series for complex numbers just like the gamma function with the facoreo?

[@drpeyam]

Not sure if a definition of odd exists for complex numbers

[@CamiKite]

Thanks for this very interesting journey! You know you did a great job when you find something that Wolfram Alpha doesn't know 😉

[@chinesecabbagefarmer]

Thanks for this upload! I'll be watching this one very SLOWLY

[@brunorepetto8928]

One of your most interesting videos. But you threw me for a spin at 10:45. You pulled the exponent 2N+1 of the -1 in series C out of a hat. But you used the correct exponent N+1 later on when you addressed series C by itself in Step 6.

[@zerospeed6412]

Why not express e^u + e^-u is 2cosh(u)? That would trigger a certain integration by parts strategy?

[@benwatson6899]

It was really an amazing integration....I loved it.....thanks a lot....

[@andygregory2390]

Fantastic when the Euler - Mascheroni Constant just vanishes. "They also serve who only stand and wait."

[@drpeyam]

Andy!!! Oh, I just made a video yesterday and thought of you, I think you’ll like it :)

[@firstlast9251]

great video as always!

[@sandorszabo2470]

I don't understand STEP 2, 4:32. 1 over 1 + exp(-2u) = 1 over 1 - exp(-2u).

[@harikrishna2k]

He corrected it...in the next line.

[@uva1312]

Great videos. I always learn a lot, keep it up.

[@thomasborgsmidt9801]

Just to nit pick, but it gives me an opportunity to demonstrate that Vivaldi and "I Lombardi" has little in common. Verdi (from "I Lomardi") with Luciano Pavarotti: kzbin.info/www/bejne/Z5zJZoqNndKUmLc The social distancing to the soprano is most probably due to Pavarotti's partiality to garlic! Let me remind You: The librettist is Temistocle Solera and not Piave. Contrast with Vivaldi (Griselda) with Cecilia Bartoli: kzbin.info/www/bejne/aICcgJmuiJWlZpo

[@andygregory2390]

Nice work again but board angle still an issue when you write on the right hand top corner

[@drpeyam]

I know... I’m filming the videos in batches, and it’s fixed in the next batch, which won’t be for another 10-20 videos or so

[@andygregory2390]

Is the tidy up at the end original to you, using reflection formula ?

[@carlosgiovanardi8197]

Awesome!! thanks for sharing.

[@6612770]

So... Are you saying that the question originally was put purely because "It should have an Answer!" ??

[@anandhuh7887]

Thanks for the video 😍

[@robertreynolds1592]

Reynolds, R.; Stauffer, A. A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function. Mathematics 2019, 7, 1148.

[@aneeshsrinivas9088]

26:45 hey don't be so mean to my friend yoshi. he freaking raised mario and is adorbs as hell.

[@robertreynolds1592]

Here is another article on ArcTangent integrals: www.mdpi.com/2227-7390/7/11/1099

[@maximilianmueller4707]

I have Kummer when See the end but still love it thank you peyam

[@drpeyam]

Hahaha, awwww!!!

[@cheshstyles]

I enjoy your videos. A suggestion: adjust your camera angle, or at least start writing on the left most side of the board. It gets a little hard to see as you work more toward the right side of the board. Just an opinion! Again, I enjoy the skill, personality and enthusiasm you bring sir :)

[@drpeyam]

Thanks!

[@MrRyanroberson1]

In the end, the poor constant was only useful as a placeholder. It got so close to being truly useful

[@ilanpi]

I. Vardi, Definite Integrals an Introduction to Analytic Number Theory, American Math. Monthly 95 (1988), 308-315.

[@TheRedfire21]

Do a derivation of the kummer series!(i dont know german :P)

[@drpeyam]

Hahaha, I actually always thought you’re German 😂 You have a very German name

[@LucaBlaLP]

I think you're missing a parenthesis in the thumbnail^^

[@drpeyam]

Oh wow, you’re right 😂

[@aneeshsrinivas9088]

lombardi? gee i've been saved by fox how swell. hey einstein i'm on your side.move it fox he's right behind you.insert some other falco quote here.

[@RaviShankar-ct7gi]

sir complex analysis questions plzz

[@drpeyam]

There’s a complex analysis playlist

[@RaviShankar-ct7gi]

sir these videos needs to be more in qwantity. only 12 . why dont you make marathon of 100 qs on complex analysis like bprp did on integrals

[@RaviShankar-ct7gi]

sir you can break bprp record

[@andriusjonaitis8509]

Sometimes I think math is like magic. :-)

[@rakhimondal5949]

Great

[@Debg91]

Kuma is bear in japanese 🐻

[@wompastompa3692]

Kuma, Chen Lu, I think Dr. Peyam likes fighting games.

[@drpeyam]

Hahahaha

[@Anokosciant]

Coomer series : *exists*

[@johannesh7610]

This Mario and Yoshi reference😁

[@stydras3380]

Nice :D

[@SartajKhan-jg3nz]

U never did the proof of A...

[@andreasxfjd4141]

Mathematica 11 cannot solve this analytically (but numerically -0.2604428...) 😌

[@lesprivatrizal]

Integral substitusi ya

[@FunctionalIntegral]

haha il tuo italiano e anche bene! :P.

[@dp121273]

Wow, this is gross!