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values of complex cosine∗
pahio†
2013-03-21 23:57:33
Since the complex cosine function z 7→ cos z has the prime period 2π, the
cosine attains all of its possible values in one of its period strips, for example in
the period strip
.
{z ∈ C .. − π 5 Re(z) < π}.
(1)
For finding out which values the cosine function can attain in a period strip, we
solve the equation cos z = w, where w is any complex number. Using Euler’s
formula
eiz + e−iz
cos z =
,
2
the equation may be written as
(eiz )2 − 2weiz + 1 = 0.
(2)
This is a quadratic equation in eiz , whence we obtain the two roots
p
eiz = w ± w2 − 1.
The product of the roots is 1, and therefore the roots are distinct from zero for
all values of w. If we set
p
w + w2 − 1 = reiϕ (−π 5 ϕ < π),
the other root is the inverse number
p
1
w − w2 − 1 = e−iϕ .
r
The solution of the equation
eiz = reiϕ
∗ hValuesOfComplexCosinei created: h2013-03-21i by: hpahioi version: h40024i Privacy
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is then obtained by taking the complex logarithm
z = z1 =
1
log(reiϕ ) = ϕ − i ln r + n · 2π
i
(n ∈ Z),
and the other solution of (2) is
z = z2 = −ϕ + i ln r + n · 2π
(n ∈ Z).
In the period strip (1), we have one solution z1 and one solution z2 , both obtained with the value n = 0 (except z2 in the case ϕ = −π with n = −1). In
(1), the points z1 and z2 are situated symmetrically with respect the origin. In
the cases w = 1 and w = −1, the equation (2) has double roots z = 0 and
z = −π, respectively; then we may say that z1 and z2 coincide. Anyhow, we
have the
Theorem. In every period strip, cosine attains any complex value at two
points.
Example. The solution of the equation cos z = 2 is obtained from eiz =
√
2± 3. In the period strip (1) we get
z =
√
√
1
log(2± 3) = −i ln(2± 3) + 0 · 2π.
i
√
Since 2± 3 are inverse numbers√of each other, we have as result the purely
imaginary numbers z = ±i ln(2+ 3).
From trigonometry, we know that the real zeros of cosine are the odd mulπ
π
π
tiples of ; from these points, ± belong to the period strip (1). Thus ± are
2
2
2
the only points of (1) where the cosine vanishes. Therefore, according to the
preceding theorem, the well-known points
(2n+1)
π
2
(n = 0, ±1, ±2, . . .)
are the only zeros of the cosine function on the whole complex plane.
The values of complex cosine function may be transferred to the complex
sine function by means of the complement formula
sin z = cos(
π
− z).
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One can think all points of the z-plane to bear the corresponding value of cosine,
and then one can translate the plane in the direction of the real axis the distance
π
; then the values of the sine have been placed to their correct places. So one
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has transferred also the above properties of cosine to sine.
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References
[1] Ernst Lindelöf: Johdatus funktioteoriaan. Second edition. Mercatorin
Kirjapaino Osakeyhtiö, Helsinki (1936).
[2] R. Nevanlinna & V. Paatero:
Otava, Helsinki (1963).
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Funktioteoria. Kustannusosakeyhtiö