Polynomial parametrisation of the canonical iterates to the solution of $-γg'= g^{-1}$

Roland Miyamoto

Polynomial parametrisation of the canonical iterates to the solution of $-γg'= g^{-1}$

Roland Miyamoto

Abstract

The iterates $h_0,h_1,h_2,\dotsc$ constructed in [8,5] and converging to the only solution $g=h\colon[0,1]\to[0,1]$ of the iterative differential equation $-γg'= g^{-1}$, $γ>0$, are parametrised by polynomials over $\Bbb Q$, and the corresponding constant $γ=κ\approx0.278877$ is estimated by rational numbers.

Polynomial parametrisation of the canonical iterates to the solution of $-γg'= g^{-1}$

Abstract

The iterates

constructed in [8,5] and converging to the only solution

of the iterative differential equation

, are parametrised by polynomials over

, and the corresponding constant

is estimated by rational numbers.

Paper Structure (3 theorems, 18 equations, 1 figure, 2 tables)

This paper contains 3 theorems, 18 equations, 1 figure, 2 tables.

Key Result

Theorem 1

Let $n\in{\mathbb N}$ and $Q_n(x):=\int_0^xq_n'q_{n-1}$ for $x\in[0,1]$. Then Wherever it makes sense, the above statements also hold for $n=0$.

Figures (1)

Figure 1: Graphs of ${h}_{0},\dots,{h}_{5}$

Theorems & Definitions (6)

Theorem 1
proof
Proposition 1
Conjecture 1
Proposition 2
proof