User talk:Ethereal kitsune

 Psi Nesting Function 

The upper and lower bounds of the index determine the degree of nesting.

1 Notation

 1.1 Algebraic Form - Radical Nesting 

an+1 = √(an) where an = a0 + √(an); where a0 is a constant

Thus,

a∞ = √(a0 + √(an)) = √(a0 + √(a0 + √(a0 + .... )))

 1.2 Expression Form 

Ψundefined (an) = Ψundefined (an + .... )1/n = (an + (an + (an + ( .... )1/n)1/n)1/n)1/n

 1.3 Function Short-hand 

Pn(x) = Ψundefined(cos(x + ...)) = cos(x + ...)

where the subscript n denotes the degree of nesting

 Examples: 

 (A) Partial Nesting 

Zeroth, First, Second, Third, N-th, ...

Rule: as with this case the assumption is that of an+1 = (an + (...)), meaning the below function is therefore only a multiple of itself. However, the second formulation shown below provides a more in-depth analysis of the Partial Psi Nesting Function.


 * Ψundefined (an) = an


 * Ψundefined (an) = (an + (an)) = 2·an


 * Ψundefined (an) = (an + (an + (an))) = 3·an


 * Ψundefined (an) = (an + (an + (an + (an)))) = 4·an


 * Ψundefined (an) = (an + (an + (an + (....)))) = n·an

 (B) Infinite Nesting 

limn→c (Ψundefined (n + .... )1/n)

= limn→e ((n + (n + (n + ( .... )1/n)1/n)1/n)1/n) = (e +(e +(e + ( .... )1/e)1/e)1/e) 1/e</sup; = √(3)

,or

limn→c (Ψundefined (a0 + .... )1/n)

= limn→2 ((a0 + (a0 + (a0 + ( .... )1/n)1/n)1/n)1/n); for a0 ≥ 0 = (2 +(2 +(2 + ( .... )1/2)1/2)1/2)1/2; where a0 = 2 = 2

,also

limn→c (Ψundefined (a0 + .... )1/n)

= limn→2 ((a0 + (a0 + (a0 + ( .... )1/n)1/n)1/n)1/n); for a0 ≥ 0 = (1 +(1 +(1 + ( .... )1/2)1/2)1/2)1/2; where a0 = 1 = φ = (1 + √(5))/2 = 1.61803398874989...