\(\sin(rx)\) 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:

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

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 

$$f^3+g^3=h^3.$$

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