278 Chapter 5y,s -1 so 81 82
i t >< t >< t
ii II
(') I (')
_.._ z __ y_ =m z + 27T- -· ------ ~ ---;:->-----
~ - \
~ \ ------
-21T -1T -'h1T 0 \ '( '121T --![...-..... 21T 37T x
'-----1T ...,. - z 1T -V--z y = -m --------
l t t t
v---- - u
Fig. 5.24A vettical line x = c -/:-^1 / 2 k7r is mapped by w = sin z onto a branch
of the hyperbola
He= {(u,v): u = sinccoshy,v = coscsinhy}where -·oo < y < +oo, or in rectangular form,
u2 v2
He: --- --=1
sin^2 c cos^2 cwhich branch (left or right) depending on whether sin c < 0 or sin c > O.
The half-line x = c, y ;::: O, with 0 < c < %7r, maps onto the upper
part of the right branch of the hyperbola He, and as y increases from 0 to+oo, that upper part is traced from the u-axis away to oo (as indicated
by the arrow in Fig. 5.24). On the other hand, the half-line x = c, y $ 0,