I'd try substituting and see where that got me. What have you tried so far?
Sorry, ignore me - I didn't notice this was a challenge forum! Okay, I'll have a go ...
Hang on, this is Bessel functions, isn't it? Way over my head, sorry about that. I'm going to have to give this one a miss till I've learned up on it.
First, use the sum of angles formula on :
so our integral becomes:
Now, one can simply note that , and that , so that our integral is
We recogize this as having the form , and so:
transgalactic, using both line integral and double integral.
Now you have got this: , can you go ahead and find
I am sure NonCommAlg has a solution for , so can you please share with us?
Additional question is: What happens if is zero or negative. Is there a closed-form solution. I guess not, if so, how about if . I don't have a solution for this. If you have, can you please share.
Let me give an example showing how the previous equality is used to get the summation of an important series.
According to the equality,
Plug (3) into (4),
Multiply on both sides,
Finally, the series:
Some Special cases:
(10) (which is within everybody's expectation)