MHF Integral Bee!
This thread is for pros! (Rock) Here is how it works: it starts with me giving you an integral to solve. If you have a complete and correct solution, post it. Otherwise do not post anything please!
So you post your solution and also your own problem (integral of course!) for others to solve. It goes on like this until this thread has 100 posts. If after 48 hours nobody solved your problem, which
is not very likely, you'll post your solution and again it'll be your turn to challenge us with a new problem. Finally whoever earns the right to post in this thread can also post his solution, if he has a
different one of course, to any problem that has already been solved in this thread. Make sure your problem is interesting and definitely not very easy to solve. Do not give any hint please!
Let's get started: Evaluate
luobo, you solved the integral i gave (good work, by the way!) but you forgot to post an integral for us to solve! (Thinking) (see the rules of this thread again!)
Sorry. See below.
Originally Posted by NonCommAlg
If is a continuous function on such that ,then evaluate
If is continous for , I think it may also satisify the identity .. because we cut the integral into many pieces and thus each improper integrals may exist .
luobo, the continuity condition for is unnecessary. i removed it from the problem. your solution is perfect! (Clapping)
simplependulum, your answer is correct and it's your turn now to challenge us with your problem! (Nod)
here's my solution: assuming that is convergent (i'll leave this open!), we have:
now put :
Here is the problem :
I've seen this one before.
Can I leave my answer in that form?
(Hi) Luobo , your solution to my problem is very cool
I want to share another method
we can consider this identity
and take the log. integral
then from L.H.S.
we can obtain
for your problem
we can change the integral in terms of gamma function:
From the Lengendre identity :