We have: for
Thus:
Integrate (assuming it's possible to exchange the series and the integral):
Now for all positive integers
Thus:
And:
Since: we get
Here is an integral I have been wrestling with. Does anyone have a technique?.
I have attempted many different ways, but always get stuck.
I have tried all sorts of series, gamma/beta, etc.
I know there is a in the solution, but getting there..
Kriz?. PaulRS?.
Commutative on SOS posed this:
Just plain ln(sinx) isn't a problem, but that x tacked on.....
I tried various series which are equaivalent:
Making a u sub gives:
I found others as well, but I will stop here for now.
I get a good ways through it, but then hit a wall.
This has been bugging me. Anyone have some clever way.
Personally, I like the gamma/beta thing, but anyway will do.
Some workings may be found here:
http://www.mathhelpforum.com/math-he...tegrals-3.html
Paul, I gotta ask. Is it that you are just naturally brilliant and just 'see' these things, derive them, know them before hand from previous applications, etc?.
For instance, how did you know to use ?.
Where does that come from?. This is certainly one wacky identity I did not know.
It amazes me how you come up with these obscure series identities.
Using the series for ln(1-x) and subbing in , I got
it down to . Which leads to
,
but got stuck there. I can't even get Maple to give me a solution of this. At least with your cos(2kx) identity it does give a closed form.
I will pick at it some more.
I am doing this to learn a little more about the way you, Kriz, PH and other gifted ones manipulate these series and identities so well.
Anyway, thanks for your input. It was great.
Yes, as a matter of fact, I did.Did you know Commutative and NonCommutativeAlgebra is the same person?
Thanks for the series derivation. You all are learning me some new things and I like that.
Opening my eyes to things I have seen but never really thought about. Stuck in a rut I suppose.