I did this quite differently. I split each integral at x=1 and used power series near zero for one piece or near infinity for the other piece, approaching x=1 in a limit. This expressed the integrals as series of the sequence s that look like (p+n)⁻¹ where p=s/k. At that point, it's just a matter of recognizing that what you get are the Eisenstein series for cotangent or cosecant, depending on if the terms alternate. You get a (-1)⁻ⁿ in the sequence for the first integral, which gives cosecant.
- I think that i wrote it wrong- B(v,y)=integral(2xcosx^v-1 sinx^y-1) from 0 to pie by tow, if the limits of integration were something else, what do we do?
I have only looked at the first integral so far. May be I have missed something. But immediately my first thought was that the integral could only exist for k sufficiently larger than s-1. Yet this does not figure anywhere in the solution. What am I missing?
"integration with respect to u" With respect to you, sir
Could you possibly think about doing a sort of “study guide” or course for integration bees?
I agree, it would be cool
Aaaaah I was waiting for that ! Thank you for it.
ptdrrr incroyable de te trouver ici
@@FroejMKmais vraiment hahaha
I did this quite differently.
I split each integral at x=1 and used power series near zero for one piece or near infinity for the other piece, approaching x=1 in a limit. This expressed the integrals as series of the sequence s that look like (p+n)⁻¹ where p=s/k.
At that point, it's just a matter of recognizing that what you get are the Eisenstein series for cotangent or cosecant, depending on if the terms alternate. You get a (-1)⁻ⁿ in the sequence for the first integral, which gives cosecant.
This is truly gorgeous! I am falling in love with the plethora that the Gamma, Beta and Digamma functions have to offer...
Axel Arno, oui monsieur ! Excellent pronunciation by the way and really nice solving ;)
Nice to have these Mellin transforms handy!
Thanks for the video, very cool integrals
These integrals converge for 0
A challenge for you 😊:
Compute limit n->infinity of
(Integral from 0 to 1 of 1/(1+x^n) dx)^n
- I think that i wrote it wrong- B(v,y)=integral(2xcosx^v-1 sinx^y-1) from 0 to pie by tow, if the limits of integration were something else, what do we do?
It is very interesting solution. Thank you
Amazing Video 😊❤
I love mathematics like you so can I talk to you please ?
Hey could you do a vedio or even a series on integrals of JEE mains or Aadvanced - i would ve intesting and informative for students
What is known about s and k in both these above?
Also, PV is due to the singularity at 1, isn't it?
Couldn’t you use ramarjun’s master theorem?
I have only looked at the first integral so far. May be I have missed something. But immediately my first thought was that the integral could only exist for k sufficiently larger than s-1. Yet this does not figure anywhere in the solution. What am I missing?
If i share this to my friends they call me math nerd
Would be good for iit adv also
Il primo è ovvio,basta porre x^k=t...si ottiene β,e poi con le proprietà di Γ,si arriva al risultato...il 2 è meno ovvio,almeno per me
Axel Arno le boss !!!
I lose a small part of my soul every time a /gamma is cancelled out from the final solution
Well technically we didn't need the digamma functions so γ was doomed from the start.