I was messing around this morning and I discovered the following, which seemed cute enough to share. In this post, I’ll make what strikes me as a very reasonable attempt to define for not an integer. Will I get the function? Wait and see!
We have . So, by basic complex analysis, , where the integral is taken along a loop around the origin. This formula is also morally right for a negative integer: wants to be $\infty$ when (because , so should be infinity, and likewise for the other negative integers). So wants to be zero for and, sure enough, this integral has no poles and vanishes in that case.
We can’t use this formula for not an integer, because has a branch cut and the path of integration would have to cross it. But we can fix that by taking the branch cut of to be along the negative real axis, and drawing our loop out to stretch very far in the negative real direction. Then will be very small at the point where the integration path crosses the real axis, so the branch cut will contribute very little. In the limit, we can define
where is a path that comes in from the negative real direction below the real axis, circles around the origin, and returns to infinity in the negative real direction above the axis. This integral will converge for all complex
So, how does this do as a definition of ? Well, it obeys the right recursion. A quick integration by parts gives , so .
Let’s take our path and shrink it towards the negative real axis. As we approach from above (for a positive real), approaches . As we approach from below, approaches . The difference between the two is . So one might think that our integral was equal to .
If you are more careful, you’ll see that this argument only works for ; otherwise, the pole at the origin is too wild to permit the limiting process. So we get that our previous definition is equivalent to
This is where a person who has seen the function defined before will say “well, you’re on the right track, but that sure looks funky.” Writing for the standard complex extension of the factorial function1, we have . So I’ve got the right integral, but it’s being evaluated at the wrong place, and there is this strange extra factor of floating around.
But it all works out! We have the functional equation of the function:
So the integral I have above really is the standard extension, but gotten at from the other side.
One wants to turn this into a proof of the functional equation, but as yet I don’t see how…