Page 28 - Calculus with Complex Numbers

Or, from the M aclaurin series we have

2 .7 A p pl ication I

W e can use the Fundamental formulae of 2.6 to obtain the real and imaginary parts
of sin z, and hence draw the graph of w = sin z. lf we write z = .x + iy, w = u + i t?
then we have

which gives

u = sin .x cosh y, t? = cos .x sinh y.

Hliminating .x we get

which is the equation of an ellipse with foci at ulu 1. Hliminating y we get

u 2 t? 2
- = 1
s in2 . x cos2 . x
which is the equation of a hyperbola with foci at ulu 1.
lt follows that w = sin z transforms the grid lines .x = constant y = constant
in the z-plane to confocal ellipses and hyperbolae in the w-plane (see Figure 2.8).
The graph of w = cos z is similar. For sinh z, cosh z we also get confocal ellipses
and hyperbolae, but with foci at Ljzi instead of at ulu 1.