1550251515-Classical_Complex_Analysis__Gonzalez_

274 Chapter^5

contained in S 0 corresponds in the w-plane the open ray
L' = {w: w = exeim = bt,O < t < +oo}
where ez = t and eim = b. As m increases from -11"+ to +11" the ray L'
turns about the origin from below the negative real axis through an angle
of 211" radians.
On the other hand, the image of the vertical segment
K = {z: z = h + iy, -oo < h < +oo, -11" < y :::;; 11"}

is the circle

K'={w: w=eheiY=reiY,-1l'<y:511"}

As y increases from -11"+ to +11", the circle of radius r is described once in

th~ positive direction. We note that the segment QP along the y-axis (for

which h = 0) is mapped onto the unit circle, that vertical segments to the

right of QP(h > 0) are mapped onto circles with radii greater than 1, while

those vertical segments to the left of QP(h < 0) are mapped onto circles
with radii less than 1. As h ---+ +oo, r = eh ---+ +oo, and as h ---+ -oo,
r ---+ 0. However, the value w = 0 is never assumed by the exponential
function (property 4). Thus we see that the function w = ez maps the
strip So onto the w-plane with the point w = 0 deleted (i.e., onto the
punctured w-plane ), and because of the periodicity, any other strip S2k1r is

mapped in the same manner. It is not possible to define conveniently ez

at oo, since lim:r; ..... 00 ez = +oo while limz->-oo ex = 0 and limy ..... 00 eiy does

not exist. Again, we may observe here the preservation of angles under the

mapping, since L( L, K) = L( L', K') = % 11".

5.19. The Circular and Hyperbolic Functions

For x real we have

- '

eix = cos x + i sin x

and

e -ix = cos x - i sin x

from which we obtain, by addition and subtraction,

cosx= ----

2

eix _ e-ix

SlnX= ----

2i

(5.19-1)

(5.19-2)

(5.19-3)

which are the celebrated formulas of L. Euler (Introduction in analysis
infinitorum, Lausanne, 1748). These formulas express the real sine and
cosine in terms of the exponential with pure imaginary exponent. Since