Is the following expression for tetration useful?

[tommy1729]

I've been playing around a bit with that integral

.

The thing is, this integral doesn't seem to converge directly. Namely, the reciprocal gamma function blows up faster-than-exponentially toward the left.

If C is allowed to be complex it seems we can do the trick :
.

This should work.

I was also thinking about using inverse Mellin too ... and then change the domain of integration ...

But this seems simpler.

However how this relates to tetration is quite another mystery to me.

regards

tommy1729