Elementary Functions
From (5.19-9) we obtainI sin zl^2 = sin^2 x cosh^2 y + cos^2 x sinh^2 y
= sin^2 x(l + sinh^2 y) + (1 - sin^2 x) sinh^2 y
= sin^2 x + sinh^2 y = cosh^2 y - cos^2 x277Hence sin z = 0 iff sin x = 0 and sinh y = O, which implies that x = br ~.
and y = 0, so that the only zeros of sin z are the real zeros z = br ( k
any integer).
Since sinhz = -isiniz, it follows that the zeros of sinhz are all of the
form iz = br or z = -bri. In a similar manner, it can be shown that the
zeros of cosz are given by z = %(2k + l)?T, while those of coshz are of
the form z = -^1 M2k + 1 )?Ti.
Next, we wish to discuss in some detail the mapping defined by w = sin z.
From (5.19-10) it follows that to the horizontal lines y = m #:: 0 correspond
the ellipses
Em= {(u,v): u = sinxcoshm, v = cosx sinhm}
where -oo < x < +oo, or in rectangular form,
u2 v2Em: 2 + 2 = 1
cosh m sinh mThese ellipses are confocal since cosh^2 m - sinh^2 m = 1, the foci being the
points w = -1 and w = +1 (Fig. 5.24). Due to the periodicity of sinx and
cos x the ellipses are described infinitely often. However, if we restrict x
to the interval [-%?T, %?T], we see from the parametric representation that
to the segment y = m > 0, -^1 / 2 ?T ::; x ::;^1 / 2 ?T corresponds the upper part
of the ellipse Em, described clockwise as x increases from -^1 / 2 7r to^1 / 2 7r,
while the lower part of the ellipse corresponds to the segment y = -m,
-%?T ::; x ::; %?T. This lower part is described clockwise if x is now
supposed to decrease from^1 / 2 7r to -^1 / 2 7r.
For y = O, -^1 / 2 7r ::; x ::; %?T, we have u = sinx, v = 0 and so the
segment [-^1 / 2 7r, 1/ 2 7r] of the x-axis corresponds the segment [-1, 1] of the
u-axis, described from -1to1 if x varies from -^1 / 2 7r to^1 / 2 7r (see Fig. 5.24).
The image of the whole real axis is again the segment [-1, 1] of the u-axis
described twice as x moves on each interval of length 2?T. t
tFor clarity, the scale on the w-plane has been made larger than the scale on
the z-plane.