Page 26 - Calculus with Complex Numbers

Ffgure 2.4

Ffgure 2.5

showing that if we use the polar form w = seis we get s = c''f
, / = y. ln other
words

Icfl = eRez rg c =
: Im c.

This will of course not be the principal value of arg ez unless -zr < lm z :jq n'.
The complex graph of w = ez is as in Figure 2.5. The grid lines .x = constant go
to circles centre the origin. The grid lines y = constant go to half lines emanating
from the origin.

2 .5 Trigonom etri c and hyperbol i c fu nctions

For real variables the trigonometric functions and the hyperbolic functions are vel'y
different animals. For example the graphs for sin .x cos .x areperiodic and bounded
(see Figtlre 2.6). W hereas the graphs for sirth .x cosh .x are neither periodic nor
bounded (see Figure 2.7).