An alternative way that uses zeta(2): After integration by parts and the substitution x=tan(z/2), the resulting integral is -int_{0}^{Pi/2} [ln(tan(z/2)]^2. Use the Fourier series expansion of ln(tan(z/2)) and orthogonality results
Since most of my exams are done (with only the Mathematics exams left), I can finally watch this big boi once more. Damn, have I missed these integrals!
This is really elegant solution! Although for me it is really difficult to come up with fourier function. A more straightfrorward way is to express last sum in terms of 2-gamma function: Sum (−1)^𝑛 /(2𝑛 + 1)^3 = (𝜓^(2) (3/4) − 𝜓^(2) (1/4))/128. And difference of 2-gamma functions comes from the third derivative of log Γ(𝑥) + log Γ(1 − 𝑥) = log 𝜋 − log sin 𝜋𝑥 at x = 3/4. 𝜓^(2) (3/4) − 𝜓^(2) (1/4) = 4𝜋^3
It was actually an exercise in my old mathematical physics textbook and I remembered it perfectly. I also remember the fourier series evaluations for zeta(2) and eta(2).
@@tahmidmosaddek6264 hit, trial and experimentation. Nothing else unless you're a genius who can do several fourier analyses in your head within seconds
@@maths_505yeah but I'm not convinced by the "boundness" argument, as the same can be argued for the function (1/x) or on the same matter, (1/x - 1) -- they would also be regarded as bounded by your argument, right? But the integral of both functions on [0; 1] is undefined, so this method fails
Hey @math505, you have so many videos relating to different concepts that Ive never heard of in calculus quite Frankly Idk where to start from because before I watched ur stuff and a few other vids I used to think I that have completed calculus and there is not much to learn other then the multivariable and Taylor and Maclaren series and stuff but boy I was wrong ....I recently discovered your channel..After seeing just the names of these concepts I feel like I've started integral calculus... I just finished area under curve and am learning Diff equations and I thought after that I completed calculus😂...Can someone take pity on my mistake soul and guild regarding what I should do after differential equations.... can I find these concepts in a book or something so that I can follow a structure along with ur videos?? pls help me math 505 I'm quite perplexed...I'm going to uni next month.... but I love calculus but it seems I have alot to learn..... I hope u see this comment...
@@Sai404wastaken honestly you won't find much of this stuff in a university calculus course. It's more like piecing together concepts from different subjects like the zeta function from analytic number theory and the series expansions are from introductory calculus. Contour integration is something you'll learn in complex analysis and you'll find laplace transforms in your course on differential equations. Just learn all that and try to apply it to integrals and it'll be a fun experience.
oh I see... I'll learn it on my own then...thx for the info btw I came across your channel as I was searching for a solution to one of mit's integration bee question... your explanation was quite good I must say...your knowledge of calculus seems very vast....thx for the videos man appreciate it alot....
Lol...... I solved this complete by seeing thumbnail only in just 2 minutes............ Coz I already knew that .. tan⁻¹x = x - x³/3 + x⁵/5 - x⁷/7 + ....... for |x|< 1 And 1/1³ - 1/3³ + 1/5³ - 1/7³ +.....= π³/32 (lnx) (tan⁻¹x)/x = -ln(1/x)(1 - x²/3 +x⁴/5 - x⁶/7 + ...... And now, just use logarithmic gamma function
How can i start study mathmatics after my 12th class ( from india) ? If anyone knows any ways please let me know.... Because I have a lot of curiosity in understanding mathmatics.....
Hey, I wanted to study theoretical physics after 12th so I applied to British unis, but I suppose u can apply to IISc, IISERs, CMI and ISI for mathematics (IISERs teach other subjects like chem and bio too, so there's that).
@@agrajyadav2951 that's a problem bro...🙂 I am a self learner And it feels so satisfying when i understand something by myself so Any reference books ?.... Btw I was thinking to give jee next year....
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Super cool! Just starting the video now. I love fourier as an audio engineer and dsp programmer.
Noice
just in time for my finishing with fourbro analysis! You have done it again my man.
An alternative way that uses zeta(2): After integration by parts and the substitution x=tan(z/2), the resulting integral is -int_{0}^{Pi/2} [ln(tan(z/2)]^2. Use the Fourier series expansion of ln(tan(z/2)) and orthogonality results
Since most of my exams are done (with only the Mathematics exams left), I can finally watch this big boi once more.
Damn, have I missed these integrals!
SUIIIIIIIIIIIIIIII
@@maths_505 SUIIIIIIIII :)
Great Job. Thank you for this smart solution.
I solved this integral using Feynman trick and reached the same result. I am really mad at you for not using the Feynman trick
This is really elegant solution! Although for me it is really difficult to come up with fourier function. A more straightfrorward way is to express last sum in terms of 2-gamma function: Sum (−1)^𝑛 /(2𝑛 + 1)^3 = (𝜓^(2) (3/4) − 𝜓^(2) (1/4))/128. And difference of 2-gamma functions comes from the third derivative of log Γ(𝑥) + log Γ(1 − 𝑥) = log 𝜋 − log sin 𝜋𝑥 at x = 3/4. 𝜓^(2) (3/4) − 𝜓^(2) (1/4) = 4𝜋^3
Amazing result! Btw how do you come up with the idea of the function to use for the Fourier expansion?
It was actually an exercise in my old mathematical physics textbook and I remembered it perfectly. I also remember the fourier series evaluations for zeta(2) and eta(2).
@@maths_505 thanks for your reply!
@@maths_505 Is there any formal way to determine the function whose fourier expansion is equal to an intended sum? besides memorizing?
@@tahmidmosaddek6264 hit, trial and experimentation. Nothing else unless you're a genius who can do several fourier analyses in your head within seconds
get well soon
Interesting that approximate value of Dirichlet function β(3)=π³/32≈0.969≈1-3%
I feel like I commented multiple times on the Fourier transform, so I kinda feel heard.
Is there any general formula for (-1)^(n)/(2n+1)^s?
UPD: Oh i see for odd s>0. It would be great to see provement
Yea the beta function of Dirichlet
Pease do more Fourier!
I guess it follows that π^3 ≈ 32. Amusingly, π^3 is very close to 31.
I didn't get it properly. why did you take f(x)=x(x-π) ?
Impressive. Who designs all these integrals and how?
I find most of them on the internet and alot of the integrals and DEs here are ones I made up myself (mostly by accident 😂)
I think it would be better to accurately use Lebesgue's Theorem of dominated convergence to swap integration and inf. summation)
Yes that is nice but lebesgue never had access to graphing software did he😂
@@maths_505yeah but I'm not convinced by the "boundness" argument, as the same can be argued for the function (1/x) or on the same matter, (1/x - 1) -- they would also be regarded as bounded by your argument, right? But the integral of both functions on [0; 1] is undefined, so this method fails
Hey @math505, you have so many videos relating to different concepts that Ive never heard of in calculus quite Frankly Idk where to start from because before I watched ur stuff and a few other vids I used to think I that have completed calculus and there is not much to learn other then the multivariable and Taylor and Maclaren series and stuff but boy I was wrong ....I recently discovered your channel..After seeing just the names of these concepts I feel like I've started integral calculus... I just finished area under curve and am learning Diff equations and I thought after that I completed calculus😂...Can someone take pity on my mistake soul and guild regarding what I should do after differential equations.... can I find these concepts in a book or something so that I can follow a structure along with ur videos?? pls help me math 505 I'm quite perplexed...I'm going to uni next month.... but I love calculus but it seems I have alot to learn..... I hope u see this comment...
mistaken*
even some of the questions that you've made videos about relating to definite integration are quite new to me tbh.... new approaches too...
@@Sai404wastaken honestly you won't find much of this stuff in a university calculus course.
It's more like piecing together concepts from different subjects like the zeta function from analytic number theory and the series expansions are from introductory calculus. Contour integration is something you'll learn in complex analysis and you'll find laplace transforms in your course on differential equations. Just learn all that and try to apply it to integrals and it'll be a fun experience.
oh I see... I'll learn it on my own then...thx for the info btw I came across your channel as I was searching for a solution to one of mit's integration bee question... your explanation was quite good I must say...your knowledge of calculus seems very vast....thx for the videos man appreciate it alot....
Questo integrale l'avevo già visto,... - B(3)...ma forse tu lo risolvi in maniera diversa
Lol......
I solved this complete by seeing thumbnail only in just 2 minutes............
Coz I already knew that ..
tan⁻¹x = x - x³/3 + x⁵/5 - x⁷/7 + ....... for |x|< 1
And 1/1³ - 1/3³ + 1/5³ - 1/7³ +.....= π³/32
(lnx) (tan⁻¹x)/x = -ln(1/x)(1 - x²/3 +x⁴/5 - x⁶/7 + ......
And now, just use logarithmic gamma function
whats gamma function for? im new to this
After which mathematics courses will I learn to solve integrals such as this? (As an undergraduate)
Just your calculus courses will be enough. Fourier series are normally taught in courses on mathematical physics.
How can i start study mathmatics after my 12th class ( from india) ?
If anyone knows any ways please let me know....
Because I have a lot of curiosity in understanding mathmatics.....
Hey, I wanted to study theoretical physics after 12th so I applied to British unis, but I suppose u can apply to IISc, IISERs, CMI and ISI for mathematics (IISERs teach other subjects like chem and bio too, so there's that).
Indian Statistical Institute, Calcutta
Did you give jee advanced??
@@agrajyadav2951 that's a problem bro...🙂
I am a self learner
And it feels so satisfying when i understand something by myself so
Any reference books ?....
Btw
I was thinking to give jee next year....
@@chaitanyasinghal1098 no...
I will next year....