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Half-iteration of x^(n^2) + 1
#1
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
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Messages In This Thread
Half-iteration of x^(n^2) + 1 - by tommy1729 - 02/13/2017, 12:12 PM
RE: Half-iteration of x^(n^2) + 1 - by Xorter - 03/04/2017, 11:43 AM
RE: Half-iteration of x^(n^2) + 1 - by tommy1729 - 03/09/2017, 01:28 PM
RE: Half-iteration of x^(n^2) + 1 - by Xorter - 03/09/2017, 10:02 PM

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