Evaluate
But without applying
Hmph. I'm not going to do it, but I wonder if you could manage to apply the definition of an integral to this? You know
The two problems I see are that depends on the integration limits, essentially forcing us to calculate , and the resulting series summation that the problem leaves could be a real bear to solve. But these problems might not be insurmountable.
-Dan
Nice problem!!!
Start by showin' that
We're going to create a double integral. (Of course, the only reason that I'd ever create a double integral in the first place is so I could reverse the integration order.) This yields
So
The right integral can be proven applyin' double integration, but I'll use another method.
Define (Note that we require
Now
Integrate for
Since we happily get
--
Yes, I used the fact that but the target was to compute the Dirichlet Integral