Genaral Pi Function and the Pi number

TL;DR

This paper introduces a new nested square root function that generates the Pi number at x=2, along with identities and asymptotic formulas related to this Pi-function.

Contribution

It defines a novel Pi-function using nested square roots and provides new identities and asymptotic formulas for it.

Findings

The Pi-function generates the Pi number at x=2.

Derived useful identities of the Pi-function.

Presented asymptotic formulas for the Pi-function.

Abstract

In this short note, we have defined a new "nested square root" function which generates usual Pi number for $x=2$. We have given some useful identities and asymptotic formulas of the Pi-function.