Complex sine and cosine function

sin z = 1/2i (e iz - e -iz) = sin x cosh y + i cos x sinh y

cos z = 1/2 (e iz +e -iz) = cos x cosh y - i sin x sinh y

= sin (x + π/2) cosh y + i cos (x + π/2) sinh y

For real numbers (y = 0 ; sinh y = 0; cosh y=1) one gets the common sine and cosine formulas. Points at the real axis are mapped periodically with period 2π into -1 ≥ sin x ≥ 1.

Points lying on parallels to the real axis are mapped to ellipses with foci +1 and -1.

Points lying on parallels to the imaginary axis are mapped to the hyperbolae with the same foci -1 and +1.

When the quadratic array is shifted parallel to the real axis its mapping will rotate once around the origin for a period of 2π of the z real axis. This strip of the z-plane is mapped to a full Riemann sheet, of which there are an infinite number. (Choose proper scaling for both planes!)

The circular array around the origin of radius π/2 is mapped to a kind of figure 8, with the accentuated points at (-1,0) and (+1,0). When it is shifted parallel to the real axis several loops may result, depending on the circle diameter. Each loop then is in a separate Riemann sheet.

The formulas show that the cosine is identical to a sine with phase shift π/2 along the real axis. This is best seen when both simulations are opened.