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[parent] values of complex cosine (Topic)

Since the complex cosine function $ z \mapsto \cos{z}$ has the prime period $ 2\pi$, the cosine attains all of its possible values in one of its period strips, for example in the period strip

$\displaystyle \{z \in \mathbb{C}\,\vdots\;\, -\pi \leqq$   Re$\displaystyle (z) < \pi\}.$ (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
$\displaystyle \cos{z} = \frac{e^{iz}+e^{-iz}}{2},$
the equation may be written as
$\displaystyle (e^{iz})^2-2we^{iz}+1 = 0.$ (2)

This is a quadratic equation in $ e^{iz}$, whence we obtain the two roots
$\displaystyle e^{iz} = w\pm\sqrt{w^2-1}.$
The product of the roots is 1, and therefore the roots are distinct from zero for all values of $ w$. If we set
$\displaystyle w+\sqrt{w^2-1} = re^{i\varphi} \quad (-\pi \leqq \varphi < \pi),$
the other root is the inverse number
$\displaystyle w-\sqrt{w^2-1} = \frac{1}{r}e^{-i\varphi}.$
The solution of the equation
$\displaystyle e^{iz} = re^{i\varphi}$
is then obtained by taking the complex logarithm
$\displaystyle z = z_1 = \frac{1}{i}\log(re^{i\varphi}) = \varphi-i\ln{r}+n\cdot2\pi \quad (n\in\mathbb{Z}),$
and the other solution of (2) is
$\displaystyle z = z_2 = -\varphi+i\ln{r}+n\cdot2\pi \quad (n\in\mathbb{Z}).$
In the period strip (1), we have one solution $ z_1$ and one solution $ z_2$, both obtained with the value $ n = 0$ (except $ z_2$ in the case $ \varphi = -\pi$ with $ n = -1$). In (1), the points $ z_1$ and $ z_2$ 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 = -\pi$, respectively; then we may say that $ z_1$ and $ z_2$ 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 $ e^{iz} = 2\!\pm\!\sqrt{3}$. In the period strip (1) we get

$\displaystyle z = \frac{1}{i}\log(2\!\pm\!\sqrt{3}) = -i\ln(2\!\pm\!\sqrt{3})+0\cdot2\pi.$
Since $ 2\!\pm\!\sqrt{3}$ are inverse numbers of each other, we have as result the purely imaginary numbers $ z = \pm{i}\ln(2\!+\!\sqrt{3})$.

From trigonometry, we know that the real zeros of cosine are the odd multiples of $ \displaystyle\frac{\pi}{2}$; from these points, $ \displaystyle\pm\frac{\pi}{2}$ belong to the period strip (1). Thus $ \displaystyle\pm\frac{\pi}{2}$ are the only points of (1) where the cosine vanishes. Therefore, according to the preceding theorem, the well-known points

$\displaystyle (2n\!+\!1)\frac{\pi}{2} \quad (n = 0,\,\pm1,\,\pm2,\,\ldots)$
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

$\displaystyle \sin{z} = \cos(\frac{\pi}{2}-z).$
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 $ \displaystyle\frac{\pi}{2}$; then the values of the sine have been placed to their correct places. So one has transferred also the above properties of cosine to sine.

Bibliography

1
ERNST LINDELÖF: Johdatus funktioteoriaan. Second edition. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1936).
2
R. NEVANLINNA & V. PAATERO: Funktioteoria. Kustannusosakeyhtiö Otava, Helsinki (1963).



"values of complex cosine" is owned by pahio. [ full author list (2) ]
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See Also: real part, properties of quadratic equation, taking square root algebraically, complex logarithm

Also defines:  period strip
Keywords:  cosine function, zeros of cosine

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Cross-references: properties, sine, distance, real axis, plane, translate, complement formula, complex sine, complex plane, vanishes, odd multiples, real, trigonometry, purely imaginary numbers, complex, origin, points, complex logarithm, solution, inverse number, product, quadratic equation, complex number, equation, cosine, prime period, function, complex cosine
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This is version 12 of values of complex cosine, born on 2007-10-31, modified 2007-12-26.
Object id is 10024, canonical name is ValuesOfComplexCosine.
Accessed 777 times total.

Classification:
AMS MSC: 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
 33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)
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