Elementary Functions 2790 < c < %7r, maps onto the lower part of the same branch, and as y
increases from -oo to 0, that lower part is traced from oo toward the u-
axis. For x = -c, 0 < c < 1/ 2 7r, u changes in sign, while v does not,
so that a similar situation arises with respect to the left branch He. The
hyperbolas obtained by varying the parameter care also confocal, with foci
at the points w = -1, and w = 1, since sin^2 c + cos^2 c = 1.
The image of the line x = 0 (the y-axis) is the line u = O, v = sinh y
(the v-axis ), the positive v-axis corresponding to the positive y-axis. More
generally, the image of any one vertical line x = n7r ( n any integer) is given
by u = O, v = (-l)nsinhy, which again defines the v-axis. What part of
the v-axis corresponds to the upper part of the line x = n7r depends on
whether n is even or odd. On the other hand, the images of the vertical
lines x =^1 M2k + l)7r are given by u = (-l)kcoshy, v = O, which define
either the infinite real interval [+1, +oo) described twice (if k is even), or
the infinite real interval (-oo, -1], also described twice (if k is odd). For
instance, as y increases from -oo to +oo along with the line x =^1 / 2 7r,
the interval [+1, +oo] is described from +oo to +1, and then a second
time from +1 to +oo, as indicated by arrows in Fig. 5.24. Similarly, as
y increases from -oo to +oo along the line x = -^1 / 2 7r, w describes the
interval (-oo, -1] from -oo to -1, then back from -1 to -oo.
From the preceding discussion it follows that the vertical strip So =
{z: -^1 / 2 7r < Rez <^1 / 2 7r,-oo < y < +oo} is mapped by w = sinz in a
one-to-one manner onto the w-plane with the real intervals [+1, +oo) and
( -oo, -1] removed, the upper part of the strip mapping onto the upper part
of the w-plane, while the lower part of So maps onto the lower w-plane. As
to the interval -%7r < x < %7r, it maps on the interval (-1, +1).
If we add to So its boundary lines, i.e., if we consider instead the strip
So= {z: -^1 / 2 7r S Rez S %7r, -oo < y < +oo}, then the mapping from So
to the w-plane ceases to be one-to-one. However, it can be made one-to-
one by making cuts along the u-axis from +1+ to +oo and from -1-to
-oo. and letting the upper edge of the cut from +1+ to +oo correspond
to the half-line x = %7r, 0 < y < +oo, and the lower edge of the cut to the
half-line x =^1 / 2 7r, -oo < y < 0. Similarly, the upper and lower half-lines
of x = -^1 / 2 7r are made to correspond to the upper and lower edges of the
cut from -1-to -oo. As to the point x = %7r, its image is +1, and that
of x = -^1 / 2 7r is -1.
Since sin(z + 2k7r) = sin z and sin( 7r - z + 2h) = sin z, all the points
z + 2k7r, as well as all the points (2k + l)7r - z, map into the same w.
Thus the strip S1 = {z: %7r < Rez < %7r, -oo < y < +oo} maps as So
does, except that its lower portion now maps onto the upper part of the
w-plane, and its upper portion onto the lower w-plane. As to the strip