280 Chapter^5
82 = {z: %rr < Rez < %rr,-oo < y < +oo} it maps exactly as So, and
so on alternatively.
Again we verify in this example that the mapping is isogonal, except at
the points z = %(2k + l)rr, where the angles are doubled.
The mapping defined by
w = cosz = sin(z + %rr)
is the same as that of w = sin z, except that any figure in the z-plane is
to undergo a preliminary translation z^1 = z + % Tr in the direction of the
positive real axis before its image under the sine function is determined.
Letting z' = iz, z^11 = ez', we may write
eiz + e-iz 1 1
w = cos z = = -(z" + - )
2 2 z"
so the mapping defined by w = cos z may also be obtained by the pre-
liminary transformations z' = iz, z^11 = ez' followed by the Joukowski
transformation.
Next, we shall discuss briefly the function
SlnZ
w=tanz= --
cosz
(5.19-11)which is defined at every point of C except at the zeros (n = %(2n + l)rr
of cos z. The points (n are said to be the poles of tan z. They are regarded
as simple poles, since the zeros of cos z are simple while sin %(2n + 1 )rr =
(-l)n. At those points we have
Because of this, the definition of tan z is completed by writing tan (n =
oo. However, the tangent function cannot be defined at oo. This point,
being an accumulation point of poles, is called a cluster point of tan z in
Chapter 9, where a more detailed study of the "singularities" of functions
is made.
As to the zeros of tan z, they are the same as those of sin z, namely, the
points en = nrr ( n = O, ±1, ±2, ... ). The function tan z is periodic, as in
the real case, with fundamental period w = rr, so thattan(z + mr) = tanz (5.19-12)
This follows at once from (5.19-11) and the properties of sinz and cosz.
Each strip