I have another point to add about Weierstrass if you guys are still considering it. I would suggest looking into the stronger, Hadamard factorization theorem.

Since we are dealing with an asymptotic half iterate of exp (I'm going to call it ) I'm going to assume we can say it's order 1?

I.e: Does for some positive M,rho?

If so then Hadamards factorization theorem guarantees that:

for some where is all the zeroes counted with multiplicity.

This also implies, quite adequately that at least for .

I think examining the zeroes is really important. I'm going to rifle through hadamard some more and see if theres anything else we can do. I believe, as tommy does, we can probably prove its order 0, so we can probably cut out and the exponential factors in the product. This would imply a very plain product.

Sorry if you guys are passed this or if it isn't useful but I realize I could've said it before, I just forgot about Hadamard ^_^. Even though its stronger and requires more work than weierstrass we only ever remember weierstrass. Poor hadamard. ^_^

Since we are dealing with an asymptotic half iterate of exp (I'm going to call it ) I'm going to assume we can say it's order 1?

I.e: Does for some positive M,rho?

If so then Hadamards factorization theorem guarantees that:

for some where is all the zeroes counted with multiplicity.

This also implies, quite adequately that at least for .

I think examining the zeroes is really important. I'm going to rifle through hadamard some more and see if theres anything else we can do. I believe, as tommy does, we can probably prove its order 0, so we can probably cut out and the exponential factors in the product. This would imply a very plain product.

Sorry if you guys are passed this or if it isn't useful but I realize I could've said it before, I just forgot about Hadamard ^_^. Even though its stronger and requires more work than weierstrass we only ever remember weierstrass. Poor hadamard. ^_^