I'm not as much of an integral guy as you are, but such elegant, meticulously crafted results do integrate the subject into a source of joy for me! Good job!
Hey man, I don't know your name, but I really appreciate all the work and care you put into your videos, they are always a nice break from my heavier university stuff. I hope you will continue your tensor series (possibly even your analytical mechanics one if that was ever going to happen). Also, could you do a video explaining when and why we can change the order of summation and integration in infinite series?
@@shivanshnigam4015 Bro I just watched some episodes and found another nice quote even regarding math: Barney: If we analyze the seemingly random patterns of the train, taking into account standard deviation, and assuming that epsilon approaches zero as angle delta approaches pi, we can conclude... Ted: [snores] Barney: Damn it, Ted! I was about to drop some sweet word play about logarithms and getting into a rhythm with my log. I'll remember it.
Since e appears within ln, it seems that this expression could be re-written without the e . So it may be artificial to claim that e is wrapped up in the solution. i.e., take the expression "ln(e)". this just equals 1, so it does not really include e .
Ah, it's always nice when the problem is actually a variation of some weird problem you solved a long time ago... The video should be called the equivalence class of the most beautiful result in calculus. Awesome work as always my friend.
I think the Integral is known as one of Malmsten's integrals. But I am not sure if it has a special name like Vardi's integral which is also one of Malmsten's. They all have really fascinating results. It's really nice that you made a video about it.
Mellin transform was always the most fun part to teach in a graduate course on Integral Transforms. You could also get the series for 1/(1 + e^-x)^2 by the binomial theorem, and then work out what the binomial coefficient (-2, k) is.
Likely a seminary trick; 2 piggybacks go through the pi hole, probably get en-natured, then are not split, but partitioned into conjugates, which produces life.
Thank you for this video. To be a complete documentation for this innovative integral, Please give the links for the proof of eta, first derivative of eta and gamma, for the non specialists and under graduate students. So, you are talented in solving such type of integrals.
That's a great result. It would be even more awesome if you didn't have to invoke "log of e^something". Do you know of an integral (or a sum) that has e, pi and gamma, but with no logarithms in sight?
I got the series by playing around with the weirstrass definition of the gamma function. I also came up with the integral (the one after my substitution) while playing around with the melin transform I derived in another video. The integral can be found in tables like the one of Prudnikov but I haven't seen the series anywhere else yet.
I've come up with quite a few of the integrals I've solved on the channel so far. Mostly by accident 😂 by messing up the solution of some other integral or experimenting with functions and limits.
Hi, I want to ask about the geometric series cant I just square the whole summation? I know its going to be impracrical but I am asking if its fine to do the squaring in normal cases. Sorry if I am fully wrong Im still learning I have to wait till high school
@@maths_505 What they mean is, you can always introduce an e into your formula, using the logarithm. The goal should therefore be to find an integral that evaluates to some combination of e, pi and gamma, but without logarithms.
You could also take 1/2 inside the logarithm to obtain a square root and a more compact result
It wouldn't be as nice though
I'm not as much of an integral guy as you are, but such elegant, meticulously crafted results do integrate the subject into a source of joy for me! Good job!
Thanks
Hey man, I don't know your name, but I really appreciate all the work and care you put into your videos, they are always a nice break from my heavier university stuff. I hope you will continue your tensor series (possibly even your analytical mechanics one if that was ever going to happen). Also, could you do a video explaining when and why we can change the order of summation and integration in infinite series?
Legen -wait for it- dary!
Barn door, Stinson natti, bro hio
@@shivanshnigam4015 nice one
@@shivanshnigam4015 Bro I just watched some episodes and found another nice quote even regarding math: Barney:
If we analyze the seemingly random patterns of the train, taking into account standard deviation, and assuming that epsilon approaches zero as angle delta approaches pi, we can conclude...
Ted: [snores]
Barney: Damn it, Ted! I was about to drop some sweet word play about logarithms and getting into a rhythm with my log. I'll remember it.
@@joniiithan yeh I've seen that too, they were in the suburbs at Lily's home 😆
Lordy. Amazing!
amazing.
Since e appears within ln, it seems that this expression could be re-written without the e . So it may be artificial to claim that e is wrapped up in the solution. i.e., take the expression "ln(e)". this just equals 1, so it does not really include e .
Well you could say a natural log is also jsut as special as e I guess.
well ln() is a function, not a special constant, and I think the author's idea is to get a conglomeration of several special constants.@@Noam_.Menashe
Ah, it's always nice when the problem is actually a variation of some weird problem you solved a long time ago... The video should be called the equivalence class of the most beautiful result in calculus. Awesome work as always my friend.
I think the Integral is known as one of Malmsten's integrals. But I am not sure if it has a special name like Vardi's integral which is also one of Malmsten's. They all have really fascinating results. It's really nice that you made a video about it.
Mellin transform was always the most fun part to teach in a graduate course on Integral Transforms. You could also get the series for 1/(1 + e^-x)^2 by the binomial theorem, and then work out what the binomial coefficient (-2, k) is.
Likely a seminary trick;
2 piggybacks go through the pi hole, probably get en-natured, then are not split, but partitioned into conjugates, which produces life.
Ok
Drop merch and I’ll be the first to get some
Thank you for this video. To be a complete documentation for this innovative integral, Please give the links for the proof of eta, first derivative of eta and gamma, for the non specialists and under graduate students. So, you are talented in solving such type of integrals.
That's a great result. It would be even more awesome if you didn't have to invoke "log of e^something". Do you know of an integral (or a sum) that has e, pi and gamma, but with no logarithms in sight?
Hey please try this one also
Integral from 0 to 1 of (x {1/x}[1/x])
Where [•] denotes floor function and {x}=x-[x]
Bro can you make a video on this one
i wonder if you could alter it slightly somehow to involve the golden ratio too
Wow now that sounds ambitious
@@maths_505 I think you can, but it will probably make it so that there isn't an Euler Mascheroni.
Did you find the series and integrals you show in your channel by yourself? Love your channel 👏🙌
I got the series by playing around with the weirstrass definition of the gamma function. I also came up with the integral (the one after my substitution) while playing around with the melin transform I derived in another video. The integral can be found in tables like the one of Prudnikov but I haven't seen the series anywhere else yet.
I've come up with quite a few of the integrals I've solved on the channel so far. Mostly by accident 😂 by messing up the solution of some other integral or experimenting with functions and limits.
@@maths_505nice, weirstrass comes up a lot between me and my students. very useful everywhere nondifferentiable function for real analysis proofs
But the eta sum in this form doesn't convergence for s=1
asnwer=1>1/2 os isit
Just integrate dx from 0 to e pi gamma
Bad ass!
Hell yeah!🔥
Good work , can you solve this integral 1/2s integral 0 to infinite of x^s/cosh x -1 to get zeta and gamma functions 😢
This is a case of Malmsten's integrals no?
The structure does agree with that
Hi, I want to ask about the geometric series cant I just square the whole summation? I know its going to be impracrical but I am asking if its fine to do the squaring in normal cases. Sorry if I am fully wrong Im still learning I have to wait till high school
Not exactly a good approach given the RHS is an infinite series.
The integral 0 to 1 of ln²(1+x)/x without Feynman integration has defeated me.
+I think the easiest way to solve it would be series expansion + cauchy's product.
It’s a fake e, cause it can be eliminated by the logarithm
Ridiculous!
e is a member of the set of real numbers!
How can you get more real than that!
@@maths_505 What they mean is, you can always introduce an e into your formula, using the logarithm. The goal should therefore be to find an integral that evaluates to some combination of e, pi and gamma, but without logarithms.
An you please take some handwriting lessons.
Nah I'm too into math to give a f**k
@@maths_505 I like your handwriting.
@@renerpho thanks bro
@rodexppi not before you take a social skills class.
+1