$x = a cosh \mu cos \nu$
$y = a sinh \mu sin \nu$

$sin z= i cos x sinh y + sin x cosh y$

$2i sinz = e^{iz}-e^{-iz}= e^{ix-y} - e^{-ix+y}$
$= e^(ix-y) - e^(ix+y) + e^(ix+y) - e^(-ix+y)$
$= e^{ix} (e^{-y} - e^y) + e^y (e^{ix} - e^{-ix})$
$= -2 e^{ix} sinh y + 2i e^y sin x$
$= -2 (cos x + i sin x) sinh y + 2i (e^y) sin x$

$sin z = i (cos x + i sin x) sinh y + (e^y) sin x$
$= i cos x sinh y - sin x sinh y + e^y sin x$
$= i cos x sinh y - sin x( sinh y - e^y)$
$= i cos x sinh y - (\frac{e^y - e^{-y} - 2e^y}{2}) sin x$
$= i cos x sinh y + (\frac{e^y + e^{-y}}{2})sin x$
$= i cos x sinh y + sin x cosh y$

$cos z= -i sin x sinh y + cos x cosh y$
$sinh z=i sin y cosh x + cos y sinh x$
$cosh z=i sin y sinh x + cos y cosh x$（这完全和椭圆坐标系对应起来了）

$sin iz=i sinh z$
$sinh iz=i sin z$
$cos iz=i cosh z$
$cosh iz= i cos z$
$|sin z|=\sqrt{sin^2 x+sinh^2 y}$
$|cos z|=\sqrt{cos^2 x+sinh^2 y}$
$|sinh z|=\sqrt{sinh^2 x+sin^2 y}$
$|cosh z|=\sqrt{cosh^2 x-sin^2 y}$

Applied complex variables for scientists and engineers