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Monday, 24 March 2025

sin(rx) for real r

\sin(nx)=\sum_{k=0}^n {n \choose k} \cos^k x \sin^{n-k} x \sin[(n-k)\frac{\pi}{2}]

Apply Newton's trick of generalizing the Bionomial expansion to real powers:

\sin(rx) :=\sum_{k=0}^\infty {r \choose k} \cos^k x \sin^{r-k} x \sin[(r-k)\frac{\pi}{2}]

Friday, 21 March 2025

Entire Dixon functions

Dixon elliptic functions \text{sm}\ z and \text{cm}\ z are meromorphic functions, both having the same poles. This implies that
\text{sm}\ z=\frac{f(z)}{h(z)}, and
\text{cm}\ z=\frac{g(z)}{h(z)}, where f, g and h are entire functions, such that 
f^3+g^3=h^3.

Tuesday, 18 March 2025

Tetration

To create tetration, it is better that we work with a commutative exponentiation

x\wedge y:=x^{\ln y}.

With this definition, tetration would be the solution to the functional equation

f(ab)=f(a)\wedge f(b)={f(a)}^{\ln f(b)}

Properties

1. f(1)=e

2. 

\ln f(\frac{1}{x})=\frac{1}{\ln f(x)},

so

3. \ln f(-1)=\pm 1