One method would be to adapt the reduction formula in this thread (modifying it so as to work for the interval instead of [0,1]).
Hi
So this is something that came from a problem we did in probability. I hope it's a correct result, since I modified it a bit.
So I know the probabilitistic way to prove it. If you can find, then do it (but you'd have to know where you're going).
Otherwise, I don't know if there is a calculus approach lol, hence this thread.
Find
for any positive integer k.
Answer should be :
Spoiler:
One method would be to adapt the reduction formula in this thread (modifying it so as to work for the interval instead of [0,1]).
another way: letteing in the beta function formula we get
now suppose and let then: therefore:
Here is another way (I'm in complex this term so here is goes)
Consider the closed arc in the complex plane. Let R be real and R > 1
and
This forms a simple closed curve in the complex plane.
By the cauchy integral formula
Now if we consider
Note that on the circular arc that
Using the ML estimate
Now if we take the limit as
We end up with