Memo: Elliptic Coordinates and Complex Trigonometric Functions

备忘：椭圆坐标与复三角函数 苏剑林 · 2011-04-10 · Mathematics coordinate ellipse trigonometric function complex number

The elliptic coordinate system is a two-dimensional orthogonal coordinate system. Its relation to Cartesian coordinates is

$$\begin{aligned}x = a \cos h \mu \cos \nu \\ y = a \sin h \mu \sin \nu\end{aligned}$$

where $(-a,0)$ and $(a,0)$ are the two foci.

See: http://zh.wikipedia.org/wiki/椭圆坐标系

Elliptical_coordinates_gridmore

Also, consider $sin z$, where $z=x +iy$. Then we have

$$\sin z= i \cos x \sin h y + \sin x \cos h y$$

Proof:

$2i sinz = e^{iz}-e^{-iz}= e^{ix-y} - e^{-ix+y}$
$$\begin{aligned}= e^(ix-y) - e^(ix+y) + e^(ix+y) - e^(-ix+y) \\ = e^{ix} (e^{-y} - e^y) + e^y (e^{ix} - e^{-ix}) \\ = -2 e^{ix} \sin h y + 2i e^y \sin x \\ = -2 (\cos x + i \sin x) \sin h y + 2i (e^y) \sin x\end{aligned}$$
Then multiplying by $-\frac{1}{2}i$:
$$\begin{aligned}\sin z = i (\cos x + i \sin x) \sin h y + (e^y) \sin x \\ = i \cos x \sin h y - \sin x \sin h y + e^y \sin x \\ = i \cos x \sin h y - \sin x( \sin h y - e^y) \\ = i \cos x \sin h y - (\frac{e^y - e^{-y} - 2e^y}{2}) \sin x \\ = i \cos x \sin h y + (\frac{e^y + e^{-y}}{2})\sin x \\ = i \cos x \sin h y + \sin x \cos h y\end{aligned}$$
Source of proof: http://au.answers.yahoo.com/question/index?qid=20100418003726AAyRTR8

There is a similar identity

$$\begin{aligned}\cos z= -i \sin x \sin h y + \cos x \cos h y \\ \sin h z=i \sin y \cos h x + \cos y \sin h x\end{aligned}$$

$cosh z=i sin y sinh x + cos y cosh x$ (this corresponds exactly to the elliptic coordinate system)

It's remarkable how closely these resemble elliptic coordinates!

There's also another interesting bit

$$\begin{aligned}\sin iz=i \sin h z \\ \sin h iz=i \sin z \\ \cos iz=i \cos h z \\ \cos h iz= i \cos z \\ |\sin z|=\sqrt{\sin^2 x+\sin h^2 y} \\ |\cos z|=\sqrt{\cos^2 x+\sin h^2 y} \\ |\sin h z|=\sqrt{\sin h^2 x+\sin^2 y} \\ |\cos h z|=\sqrt{\cos h^2 x-\sin^2 y}\end{aligned}$$

Reference: Applied complex variables for scientists and engineers