Total Internal Reflection for Electron waves

Idea: Hamilton's opto-mechanical analogy makes it plausible that Snell's law can apply to `rays of electrons'.

Application: Build waveguides for electrons based on total internal reflection of de Broglie matter waves

\(\sin(rx)\) 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 

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