Vardi Integral

Рет қаралды 14,845

5 жыл бұрын

In this video, I evaluate the Vardi integral, which is the integral from pi/4 to pi/2 of ln(ln(tan(x)). Although the calculation is quite involved, it's really exciting, and brings us to the world of gamma functions, the Kummer series, and the Euler-mascheroni constant! Enjoy!

Пікірлер: 81

@weinihao3632 5 жыл бұрын

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 5 жыл бұрын

That is amazing, thank you for the cool comment!

@rot6015 5 жыл бұрын

this is awesome, thanks for sharing

@saulmendoza1652 5 жыл бұрын

Nice publication.

@cycklist 5 жыл бұрын

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

@pablojulianjimenezcano4362 5 жыл бұрын

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

@robertreynolds1592 4 жыл бұрын

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.

@emanuelmartinez3585 5 жыл бұрын

More series and integrals! OMG So much fun 🙊

5 жыл бұрын

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

@Rundas69420 5 жыл бұрын

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

@CamiKite 5 жыл бұрын

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

@chinesecabbagefarmer 5 жыл бұрын

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

@uva1312 5 жыл бұрын

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

@neilgerace355 5 жыл бұрын

1:20 Unleash the power of CHEN LU!

@chandankar5032 5 жыл бұрын

Wow ! Seems like you are after flammable maths

@firstlast9251 5 жыл бұрын

great video as always!

@carlosgiovanardi8197 4 жыл бұрын

Awesome!! thanks for sharing.

@DevalMehtaAstrokidintraining 5 жыл бұрын

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).

@benwatson6899 5 жыл бұрын

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

@GreenMeansGOF 5 жыл бұрын

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

@anandhuh7887 5 жыл бұрын

Thanks for the video 😍

@karolakkolo123 5 жыл бұрын

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 5 жыл бұрын

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

@FreeGroup22 3 жыл бұрын

Did you find it?

@karolakkolo123 3 жыл бұрын

@@FreeGroup22 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

@pedroalonso7606 3 жыл бұрын

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.

@brunorepetto8928 5 жыл бұрын

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.

@andygregory2390 5 жыл бұрын

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

@drpeyam 5 жыл бұрын

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

@albertemc2stein290 5 жыл бұрын

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 5 жыл бұрын

It’s an improper integral :)

@albertemc2stein290 5 жыл бұрын

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

@quantumcity6679 5 жыл бұрын

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 5 жыл бұрын

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 5 жыл бұрын

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

@juauke 5 жыл бұрын

@@quantumcity6679 you're welcome :^)

@quantumcity6679 5 жыл бұрын

@@juauke appreciate... 😅

@rakhimondal5949 5 жыл бұрын

Great

@maximilianmueller4707 5 жыл бұрын

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

@drpeyam 5 жыл бұрын

Hahaha, awwww!!!

@lucasdepetris5896 5 жыл бұрын

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 5 жыл бұрын

Not sure if a definition of odd exists for complex numbers

@aneeshsrinivas9088 2 жыл бұрын

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

@zerospeed6412 5 жыл бұрын

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

@stydras3380 5 жыл бұрын

Nice :D

@robertreynolds1592 4 жыл бұрын

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

@thomasborgsmidt9801 3 жыл бұрын

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: kzfaq.info/get/bejne/Z9GWaLqIzdPFkok.html 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: kzfaq.info/get/bejne/aLVpgsmpuJbWYGw.html

@6612770 5 жыл бұрын

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

@andriusjonaitis8509 5 жыл бұрын

Sometimes I think math is like magic. :-)

@ilanpi 4 жыл бұрын

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

@MrRyanroberson1 5 жыл бұрын

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

@aneeshsrinivas9088 2 жыл бұрын

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.

@cheshstyles 5 жыл бұрын

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 5 жыл бұрын

Thanks!

@andygregory2390 5 жыл бұрын

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

@drpeyam 5 жыл бұрын

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 5 жыл бұрын

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

@TheRedfire21 5 жыл бұрын

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

@drpeyam 5 жыл бұрын

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

@johannesh7610 5 жыл бұрын

This Mario and Yoshi reference😁

@robertreynolds1592 4 жыл бұрын

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

@FreeGroup22 3 жыл бұрын

Coomer series : *exists*

@LucaBlaLP 5 жыл бұрын

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

@drpeyam 5 жыл бұрын

Oh wow, you’re right 😂

@sandorszabo2470 5 жыл бұрын

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

@harikrishna2k 5 жыл бұрын

He corrected it...in the next line.

@Debg91 5 жыл бұрын

Kuma is bear in japanese 🐻

@wompastompa3692 5 жыл бұрын

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

@drpeyam 5 жыл бұрын

Hahahaha

@RaviShankar-ct7gi 5 жыл бұрын

sir complex analysis questions plzz

@drpeyam 5 жыл бұрын

There’s a complex analysis playlist

@RaviShankar-ct7gi 5 жыл бұрын

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 5 жыл бұрын

sir you can break bprp record

@FunctionalIntegral 5 жыл бұрын

haha il tuo italiano e anche bene! :P.

@andreasxfjd4141 5 жыл бұрын

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

@lesprivatrizal 4 жыл бұрын

Integral substitusi ya

@SartajKhan-jg3nz 5 жыл бұрын

U never did the proof of A...

@dp121273 5 жыл бұрын

Wow, this is gross!