Elementary Functions 281is called a period strip for w = tan z. In particular,So= {z: -%rr < Rez::::;^1 / 2 rr,-oo < Imz < +oo}
will be called the fundamental period strip. From the additional theorems
of sine and cosine it follows, as in ordinary trigonometry, that( )tanz 1 + tanz2
tan z1 + z2 = ------
1 - tanz1 tanz2
whenever z1, z 2 , and z 1 + z 2 are different from %(2n + l)rr.
In particular, for z f= %(2n + l)rr we have. tan x + i tanh y
tan z = tan( x + zy) = l. h
- z tanxtan y
tan x(l -tanh^2 y). (1 + tan^2 x) tanh y
= 2 + z 2
1 + tan^2 xtanh y 1 + tan^2 xtanh y(5.19-13)(5.19-14)Hence tan z is real iffy = 0 (i.e., if z is real), and tan z is pure imaginaryon the vertical lines z = nrr + iy, -oo < y < +oo. Also, tanz is pure
imaginary on the vertical lines z =^1 / 2 (2n + l)rr + iy, y f= 0. To see this,
recall that tan z in terms of the exponential is given by
e2iz _ 1
tanz = -i- 2 .--
e iz + 1For z =^1 /i2n + l)rr + iy, we have
so we obtaine2iz = e-2y+(2n+l)1l'i = -e-2y
1 + e-^2 v
tan[^1 /i2n + l)rr + iy] = i
1_ e- 2 y
which is pure imaginary if y f= ·o. Note that (5.19-16) gives
y-->+oo lim tan[%(2n + l)rr + iy] = i
and
lim tan[%(2n + l)rr + iy] = - i
y-->-oo(5.19-15)(5.19-16)The same property holds as y -+ +oo or as y -+ -oo along any other
vertical line z = x 0 + iy, as can be seen from (5.19-14), since tanhy-+ 1
as y-+ +oo while tanhy-+ -1 as y-+ -oo. The values w = ±i are not
actually assumed by tan z for any z E C. In fact, the equations
e2iz _ 1
tan z = -i. = ±i
e2iz + 1