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[parent] periodicity of exponential function (Theorem)
Theorem 1   The only periods of the complex exponential function $ z\mapsto e^z$ are the multiples of $ 2\pi i$. Thus the function is one-periodic.

Proof. Let $ \omega$ be any period of the exponential function, i.e. $ e^{z+\omega} = e^ze^\omega = e^z$ for all $ z\in\mathbb{C}$. Because $ e^z$ is always $ \neq 0$, we have

$\displaystyle e^\omega = 1.$ (1)

If we set $ \omega := a+ib$ with $ a$ and $ b$ reals, (1) gets the form
$\displaystyle e^a\cos{b}+ie^a\sin{b} = 1,$ (2)

which implies (see equality of complex numbers)
$\displaystyle e^a\cos{b} = 1,\quad e^a\sin{b} = 0.$
As these equations are squared and added, we obtain $ e^{2a} = 1$ which means, since $ a$ is real, that $ a = 0$. Thus the preceding equations get the form
$\displaystyle \cos{b} = 1,\quad \sin{b} = 0.$
These result that $ b = n\cdot 2\pi$ and therefore
$\displaystyle \omega = n\cdot 2\pi i\quad (n \,=\, 0,\,\pm 1,\,\pm 2,\,\pm 3,\,\ldots)$
Q.E.D.

Bibliography

1
ERNST LINDELÖF: Johdatus funktioteoriaan (`Introduction to function theory'). Mercatorin kirjapaino, Helsinki (1936).



"periodicity of exponential function" is owned by pahio.
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See Also: periodic functions, analytic continuation of Riemann zeta (using integral), examples of periodic functions

Also defines:  one-periodic

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Cross-references: equations, equality of complex numbers, implies, reals, exponential function, proof, function, multiples, complex exponential function, periods
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This is version 10 of periodicity of exponential function, born on 2005-05-23, modified 2008-05-02.
Object id is 7107, canonical name is PeriodicityOfExponentialFunction.
Accessed 2828 times total.

Classification:
AMS MSC30D20 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Entire functions, general theory)
 32A05 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Power series, series of functions)

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