02/13/2017, 12:12 PM

I was intrested in the half-iteration of f_n(x) = x^(n^2) + 1 for Large x.

For instance g_n(x) = f_n ^[1/2](x) - x^n.

H_n(x) = f_n ^[1/2](x) / x^n.

For Large x :

Is abs g_n(x) increasing or decreasing with n ?

Is abs H_n(x) decreasing ?

Probably abs g_n is increasing and abs H_n decreasing.

The focus is on integer n and branch structure.

But also if n is real , are these functions analytic in n ?

Perturbation Theory suggests this.

I wonder how these functions look like on the complex plane , especially with resp to n.

Regards

Tommy1729

For instance g_n(x) = f_n ^[1/2](x) - x^n.

H_n(x) = f_n ^[1/2](x) / x^n.

For Large x :

Is abs g_n(x) increasing or decreasing with n ?

Is abs H_n(x) decreasing ?

Probably abs g_n is increasing and abs H_n decreasing.

The focus is on integer n and branch structure.

But also if n is real , are these functions analytic in n ?

Perturbation Theory suggests this.

I wonder how these functions look like on the complex plane , especially with resp to n.

Regards

Tommy1729