The Behaviour of sin z
In these notes we look at the values that sin z takes as z runs over C. We recall that
sin z = sin(x + iy) = sin x cos(iy) + cos x sin(iy)
= sin x cosh y + i cos x sinh y
where
ey − e−y
ey + e−y
cosh y =
2
2
Here are the properties of sinh y and cosh y that we are going to use:
◦ When y = 0, we have sinh y = 0, cosh y = 1.
◦ When y → +∞, we have sinh y ≈ cosh y ≈ 12 ey → ∞.
◦ When y → −∞, we have sinh y ≈ − 12 e−y → −∞ and cosh y ≈ 21 e−y → +∞.
◦ sinh(−y) = − sinh y and cosh(−y) = cosh y.
◦ sinh y is positive for y > 0 and negative for y < 0.
◦ cosh y ≥ 1, since cosh 0 = 1, ddy cosh y = sinh y is positive for y > 0, and cosh y is even.
sinh y =
2
For small y, cosh y = 1 + y2 + · · ·, where the · · · refers to higher powers of y. To see
2
2
this, just substitute ey = 1 + y + y2 + · · · and e−y = 1 − y + y2 + · · · into the definition
of cosh y.
◦ cosh2 y − sinh2 y = 1.
The real and imaginary parts of sin z are
u = sin x cosh y
v = cos x sinh y
So
◦ If y is held fixed at y = 0, then v = 0 and u = sin x runs between −1 and +1.
◦ If y is held fixed at some nonzero value, then u = sin x cosh y and v = cos x sinh y lie
u2
v2
on the ellipse cosh
2 y + sinh2 y = 1. For y small, the semimajor axis cosh y, which is
in the u direction, is just a little larger than 1 and the semiminor axis | sinh y|, which
is in the v direction, is just a little larger than 0. For y large, the semimajor and
semiminor axes are both large with the former just a little larger than the latter.
◦ If x is held fixed at x = kπ, with k integer, then u = 0 and v = (−1)k sinh y runs
between −∞ and +∞.
◦ If x is held fixed at x = k + 21 π, with k integer, then v = 0 and u = (−1)k cosh y
runs between 1 and +∞ for k even and between −1 and −∞ for k odd.
◦ If x is held fixed at some value which is not an integer or half integer multiple of π,
2
2
then u = sin x cosh y and v = cos x sinh y lie on the hyperbola sinu2 x − cosv 2 x = 1.
These are sketched in the figure below.
c Joel Feldman.
2012. All rights reserved.
January 24, 2012
The Behaviour of sin z
1
v
x = kπ
x = −0.1π
y=2
x = 0.1π
x = −0.4π
x = 0.4π
y=
x = −π/2
1
4
y=0
x = π/2
u
c Joel Feldman.
2012. All rights reserved.
January 24, 2012
The Behaviour of sin z
2