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Gamma Digamma
India
Приєднався 27 вер 2015
Here I will do mathematics: from challenging integrals, infinite series, linear algebra, etc. and physics.
I also have a playlist for my piano performances that I keep updating from time to time.
Subscribe to my channel to learn something new, using the very foundations that our universe is built off of.
Remember, knowing your "limits", you are limitless...
I also have a playlist for my piano performances that I keep updating from time to time.
Subscribe to my channel to learn something new, using the very foundations that our universe is built off of.
Remember, knowing your "limits", you are limitless...
Second Barnes Lemma
In this video, which is a continuation of the last one, I state and prove the Second Barnes Lemma.
#gammafunction #calculus #integrationcalculus #integration #definiteintegral #complexanalysis #laplacetransform #fouriertransform #mellintransform #mellinbarnes #barnesfirstlemma #convergence #poles #residuetheorem #stirlingsformula #exponential #quantumfieldtheory #feynmandiagrams #feynmanpropagator #highenergyphysics #barnessecondlemma #radiusofconvergence #nestedintegrals
#drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stevemould #vibingmaths
#gammafunction #calculus #integrationcalculus #integration #definiteintegral #complexanalysis #laplacetransform #fouriertransform #mellintransform #mellinbarnes #barnesfirstlemma #convergence #poles #residuetheorem #stirlingsformula #exponential #quantumfieldtheory #feynmandiagrams #feynmanpropagator #highenergyphysics #barnessecondlemma #radiusofconvergence #nestedintegrals
#drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stevemould #vibingmaths
Переглядів: 80
Відео
Mellin transform, Mellin-Barnes Integral and First Barnes Lemma
Переглядів 13614 годин тому
In this content-heavy video, I first define the Mellin transform and its inverse. I then use this to prove a famous integral, the Mellin-Barnes Integral, which has applications in quantum field theory. Finally, I use this integral to prove the uncanny First Barnes Lemma. #gammafunction #calculus #integrationcalculus #integration #definiteintegral #complexanalysis #laplacetransform #fouriertrans...
Computing the Volume and Surface Area of a Unit Sphere in n-dimensions
Переглядів 124День тому
In this video, as the title suggests, I am going to compute the volume and surface area of a unit sphere in n-dimensions and then end the video by giving an application in electromagnetism. #sphere #integral #integralcalculus #dimensions #polar #variables #substitution #gammafunction #recursion #series #shells #algebra #legendre #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn...
Gauss Multiplication Formula for the Gamma function
Переглядів 14114 днів тому
In this video, I prove the Gauss multiplication formula for the Gamma function using Stirling's approximation from the last video. #gauss #gammafunction #multiplication #limits #calculus #asymptotic #product #series #arithmeticseries #integers #recursion #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stevemould #vibingmaths
An Elementary Proof of Stirling's Formula
Переглядів 15421 день тому
In this video, I prove the Stirling's formula, which has several applications in statistics, thermodynamics and probability, using elementary techniques. #stirlingsformula #stirling #sequences #realanalysis #analysis #calculus #logarithm #exponential #gammafunction #factorials #limit #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stevemoul...
Infinite Product Representation Of The Sine
Переглядів 14421 день тому
In this video, I present a proof of the infinite product representation of the sine function i.e. the Wallis product. #sines #trigonometry #product #limit #calculus #derivatives #infinite #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stevemould #vibingmaths
Infinite Sum of 1/n^2 : Cauchy's Proof
Переглядів 17028 днів тому
In this video, I solve the Basel problem, evaluating the infinite sum of 1/n^2, using one of the most elegant proofs presented by Cauchy. #basel #infinitesum #sum #infinity #pi #cotangent #trigonometry #sines #cosine #tangents #polynomials #vietastheorem #roots #integer #euler #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stevemould #vibi...
Fascinating Log Contour Integral
Переглядів 380Місяць тому
In this video, I am evaluating a fascinating integral using contour integration about the poles of the logarithm function in the complex plane, ending up with two improper integral for the price of one. #residuetheorem #complexanalysis #integration #integralcalculus #definiteintegrals #improperintegrals #calculus #poles #logarithmicfunctions #branchcuts #phases #contourintegration #drpeyam #bla...
Mystical Generalization Of The Fractional Part Integral
Переглядів 203Місяць тому
In this video, I evaluate a fascinating generaliztion of the fractional part integral on n-dimensions with a very satisfying result. #integralcalculus #definiteintegral #improperintegrals #fractionalpart #floorfunction #infinty #substitution #byparts #eulermascheroni #logarithmicfunctions #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stev...
Insane Integral With A Beautiful Result
Переглядів 607Місяць тому
In this video, we evaluate a crazy looking integral to get a very beautiful and satisfying result. #integration #calculus #integralcalculus #definiteintegral #improperintegrals #infinity #gammafunction #digamma #polygamma #specialfunctions #sines #trigonometry #feynmantechnique #leibnitz #derivatives #differentialcalculus #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khana...
Pendulum Without The Small Angle Approximation
Переглядів 954Місяць тому
In this video, I solve the non-linear dynamics problem of the pendulum without using the small angle approximation. The final result involves elliptic integral functions and beta functions. Intro: (00:00) First integral of motion: (00:40) Consequences: (10:06) General solution: (13:40) Time period: (21:00) Small angle limit: (29:20) #physics #pendulum #smallangle #mechanics #mathematics #calcul...
Symmetric Double Integral
Переглядів 2172 місяці тому
In this video, I evaluate a definite double integral with a symmetric integrand over a 1×1 square region. After tending to the limiting case as n goes to ∞, I evaluate another similar integral yielding a nice result. #doubleintegrals #multivariablecalculus #calculus #square #maths #integralcalculus #symmetry #graphs #substitution #area #drpeyam #blackpenredpen #flammablemaths #3blue1brown #mich...
Integral of the exponential of cosines
Переглядів 3854 місяці тому
In this video, I evaluate two simple looking yet interesting definite integrals involving the exponential of cosines. #exponential #integral #definiteintegrals #cosine #sine #trigonometry #calculus #sum #infinite #residue #complexnumbers #contour #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #vibingmaths
Two mathematical approaches and one physical application
Переглядів 2504 місяці тому
In this video, I evaluate a definite integral in two different ways and then apply it to find the electric field of a continuous charge distribution. Intro (00:00) First approach (00:42) Second approach (10:30) Physical application (19:23) #integration #definiteintegrals #pi #triangle #cosine #vectors #lawofcosines #theorems #parallelogram #substitution #electromaganetism #electricfield #electr...
Napoleon's Theorem
Переглядів 2654 місяці тому
In this video, I will be stating and proving Napoleon's theorem using complex numbers. Featuring the cube roots of unity. #napoleon #theorem #geometry #algebra #complexnumbers #triangle #equilateraltriangle #trigonometry #rotation #realnumbers #imaginarynumbers #proof #centroid #centroidoftriangle #median #perpendicular #drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanaca...
Anti-discrimination, anti-racism: an Indian's perspective
Переглядів 370Рік тому
Anti-discrimination, anti-racism: an Indian's perspective
Complete Elliptic Integral of the First Kind
Переглядів 3,5 тис.Рік тому
Complete Elliptic Integral of the First Kind
Triple Fractional Part Impossible Integral
Переглядів 550Рік тому
Triple Fractional Part Impossible Integral
Generalized "Powerful" Fractional Part Integral
Переглядів 301Рік тому
Generalized "Powerful" Fractional Part Integral
A New Proof of Pythagoras' Theorem feat. Law of Sines
Переглядів 317Рік тому
A New Proof of Pythagoras' Theorem feat. Law of Sines
CFRCE Lecture: Introduction to Constrained Dynamics 2
Переглядів 316Рік тому
CFRCE Lecture: Introduction to Constrained Dynamics 2
CFRCE Lecture: Introduction to Constrained Dynamics 1
Переглядів 213Рік тому
CFRCE Lecture: Introduction to Constrained Dynamics 1
Good, Bad and Ugly Constants feat. Balanced Polygamma
Переглядів 5462 роки тому
Good, Bad and Ugly Constants feat. Balanced Polygamma
Thank you for the explanation.. but i have a question, isn't this integral an improper integral since the function is not defined when x=0 which is the lower bound?
The integration interval is (0, pi/2], so zero is excluded for the integrand to be well-defined. So, yes it is an improper integral.
Good topic. Always wanted to learn this.
There is a typo in the sum introduced at 10:52 until 12:09, it should be k instead of n as the summing variable.
The sums are from k equal one to infinity. Be careful with the index used in your sums.
Yes, it is a typo, it should be from k and not n. The sum is still from 0 to infinity though, since I have taken 1/(k+1) as the first term of the summand.
One of my favorite topics. Fun fun fun. There's a good intro to the Mellin transform in "Integral Applications and Their Applications" by Davies , a book that I've used over the years in graduate courses at Georgia Tech. But of course, the really fun part is Ramanujan's Mater Theorem.
I apologise if I made any mistakes or misspoke at any time in this video. The content of this video was comparatively quite involved.
Hi, sum (-1)^(k+1)*k^2/(k^3+1), k=1 to inf
Hi, thanks for your suggestion but I don’t think it has a closed form answer.
You always have interesting ideas for videos. If an idea for an interesting topic comes to me, I'll try to run it by you.
Thank you. Yes, I would appreciate that 😊
Use floor functions for those indexes in your binomial expansion at the start.
You are right, thanks. I should have also specified that m is an integer.
Your work is way too good. I’ve been studying special functions for a long time. Your derivations are clean, complete and exhaustive. You are a natural teacher explaining hard concepts. Keep working hard, you are going up.
Is a Trigonometric Proof Possible for the Theorem of Pythagoras? Michael de Villiers RUMEUS, University of Stellenbosch CONCLUDING COMMENTS To get back to the original question of whether a trigonometric proof for the theorem of Pythagoras is possible, the answer is unfortunately twofold: yes and no. 1) Yes, if we restrict the domain to positive acute angles, any valid similarity proof can be translated into a corresponding trigonometric one, or alternatively, we could use an approach like that of Zimba (2009) or Luzia (2015). 2) No, if we strictly adhere to the unit circle definitions of the trigonometric ratios as analytic functions, since that would lead to a circularity.
Use the reimann zeta function plug In 2 get the answer
And this is how you prove that the Riemann Zeta of 2 is pi^2/6
Love this channel.
I love your videos, dude. You deserve more subs and views but I appreciate you not resorting to adopting a 'UA-cam personality' or trying to make click bait thumbnails for your videos (if those like sound like wild approaches to a math channel, well you'd be right, but that doesn't mean it's not ubiquitous). Keep up the great videos. What's next on the upload agenda, or is that a surprise?
Thanks, I appreciate it. I was thinking of doing some more integrals, do you have any particular recommendations regarding topics in mathematics/physics? Once I have more time on my hands, maybe I could introduce Lagrangian and Hamiltonian mechanics via variational calculus.
Your explanation is very nice. At k=1 what happened for complete elliptic integrals. Whether it has logarithmic singularities at k=1? I
When k=1, you can use trigonometric substitution to solve the elementary definite integral.
Bro where are you from?
India. What about you?
@@Gamma_Digamma I know India. But from which state?
@@Gamma_Digamma I'm from West Bengal..
@@Pritam_Jana_uMaharashtra
@@Gamma_Digamma I have a integration problem can I send you that?
Thank you, sir! 🎉 "Knowing your limits, you are limitless."
Well presented. Articulate ❤
Super
So elegant , so beautiful, just looking like a WOW! I have to say level of rigour in this lecture series is rather impressive and the clarity is top notch, thank you so much for sharing as this is rather a niche topic which is not covered in conventional textbooks!
Thanks a lot 🙏
This video is so great, i wonder why it only got a few visualizations... And also I'm FIRST to comment LESSSSS GOOOOOOOOOOO
Pronunciation: it is Integral, not Inteegral!
i know you found a way around it, but as a beginner in complex analysis i wonder, isn't the ln function defined on the complex plane except the negative real axis? Thanks anyways, well done.
no, e^ipi is -1 so ln(-1) is perfectly defined, and is just ipi(1+2n) for integers n, depending on which branch you pick
@@mokshnihalani1679Yes, precisely.
Interesting! I'm sure it's not very significant, but it's amusing that the value of the integral for a, b > 0 is the product of the *logarithm of the Geometric Mean of a and b* and the *Logarithmic Mean of a and b* . Also, if a=b=c>0, then the maximum value for I occurs when c=e, with the integral of log(x) / (x+e)² from 0 to ∞ evaluating to e. If one of {a,b} is equal to 1, then the maximum value of I occurs when the other parameter equals exp(2 + W₀(- 2 e⁻²)) ≈ 4.9215536 where W₀ is the principal branch of the Lambert W aka "product log" function. Thanks for the video. I'm currently fascinated by functions involving products of logs and/or logs raised to powers. Complex analysis is so mysterious and somewhat unintuitive for me, but very powerful!
Fixing one of the {a,b} to be 1, you can get much greater values for the integral if the other parameter is allowed to be negative (assuming we take the Cauchy Principal Value of the integral on (0,∞) ). A couple of neat values: With a = 1, bn= -1, we get a nice I = π²/4. For a = 1, b = -2, we get I = ⅙ (π² - log²(2)). Edit: Sorry, in my enthusiasm I forgot that you show the π²/4 result in the video 😅
It’s certainly a very good observation and certainly not insignificant. Thanks for your comment :)
@@Gamma_Digamma sorry that I repeated a result from your video in my next comment! I made an edit and mentioned that you cover the log(x) / (x²-1) integral in your video! 😆
@@knivesoutcatchdamouse2137no worries
Nice 🔥
just quite tedious to calculate but it's pretty easy in itself, when you had the integral from zero to infinity of x^(k-1)/(1+x^2) you could have noticed that it's a quite known mellin transform, which instantly evaluates to pi/2*csc(kpi/2) nice problem
My subscribers would then have requested that I explicitly define and calculate/derive the Mellin transform formula you are referring to. Better to do it in one shot with already established facts than looking up values in tables.
Thank you very very nice
impressive, and thanks for showing how the general case translates to the small angle approximation as a first order term
hi sir can you drive this functional equation using the hankel contour
I'll see what I can do...
32:53 "A real valued function" Isn't f(x) still complex valued even when you only consider real x? The original integrand for the complex analysis step is at (16:18): f(x) = [(x-i)ln(x+i) - (x+i)ln(x-i)]/[2i(x^2+1)] I don't think that that's real valued on x in [0, infinity). That's the only thing I'm hung up on. Otherwise, this is absolutely awesome!
You make a good point. I think I misspoke at that part. I urge you to check out this revisit video that has me do the complex analysis more rigorously, involving log branch cuts: ua-cam.com/video/6Sefmkv2gUo/v-deo.htmlsi=aaO30gRwGr2EvHQg
Leaving this comment so I will remember to come back
Same here
Thanks for explaining the graphical part. I had doubt in that part, but now it is clear.
At 1:10 if you make the substitution x = pi/2 - x, then dx = -dx, but you didn't reflect that.
The minus sign resulting from the dx was used to flip the bounds of integration which were going from pi/2 to 0 as a result of the substitution. I have directly applied King's property.
@@user-pt1ky6db9yno problem 😊
Loved the video. Keep up the good work and congrats on your graduation 🎉
Nice format
I am really unsure whether infinity in the integration limits can be replaced by n . The reason i am unsure is not all infinities are same. I mean i can write as lim m approaches inf , lim n approaches inf integral 0to m (1-t/n)^n * t^(x-1) dt but m can be replaced by n if and only if it is known that m-n or n-m is a finite quantity
Gamma (x ) obv
You are right, I have not been rigorous with my derivation. The reason why this works is similar to that of Cauchy principal values.
nice tutorial, thank you sir
Excellent video!
This is actually really comprehensive. Thanks
If you replace the cosine by a sine in the same integral going from -pi to pi, you will get the same result as with the cosine. We see this using the change of variables x-->pi/2-x and the fact that we are integrating in a full circle (0 to 2pi).
What justifies being able to ignore Cn for n=0 simply because it diverges?
If you try to calculate c_0 explicitly, you indeed get zero. I believe I miswrote the Fourier series expansion when I summed from -infinity to infinity; since c_0 is zero, the expansion is actually from -infinity to -1 and from 1 to infinity. Sorry for the lazy arguments, this is an older video.
I like the title. The presentation is neat as well.
I am trying to learn the way you show us here... Can I say that point H is 2z - 2y times a rotation of 30 degrees times a magnitude correction so it is 2(z-y) time 3^0.5 / 2 times -1 times (3^0.5 /2 + 0.5 i ) ? or this is "only" CH vector ?
Since you are rotating vector CA by 30 degrees, what you describe is only vector HC. You will have to add C to HC to obtain point H.
Thank you bro!!! Very excellent solve!