Integral involving expoential - does it have a name?

I'm working with this simple little integral,

,

and was wondering if it had a name. It seems so basic that I imagine there's a lot of literature written about it - I would like to know more about it.

An observation: *If* the limits were 0 and infinity the solution would be related to gamma functions.

I have found solutions for integer (integrate by parts once, then note a recursion relation). Then general solutions can be written down; one for even and one for odd .

e.g. for odd

I think that's correct - I'm just getting the feeling that it must have been solved a million times before...

Any thoughts?

Thanks in advance,

Tom.