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periodicity of exponential function
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(Theorem)
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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
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(1) |
If we set $\omega := a+ib$ with $a$ and $b$ reals, (1) gets the form
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which implies (see equality of complex numbers) $$e^a\cos{b} = 1,\quad e^a\sin{b} = 0.$$ 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 $$\cos{b} = 1,\quad \sin{b} = 0.$$ These result that $b = n\cdot 2\pi$ and therefore $$\omega = n\cdot 2\pi i\quad (n \,=\, 0,\,\pm 1,\,\pm 2,\,\pm 3,\,\ldots)$$ Q.E.D.
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- ERNST LINDELÖF: Johdatus funktioteoriaan (`Introduction to function theory'). Mercatorin kirjapaino, Helsinki (1936).
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"periodicity of exponential function" is owned by pahio.
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Cross-references: equations, equality of complex numbers, implies, reals, exponential function, proof, function, multiples, complex exponential function, periods
There are 4 references to this entry.
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 3651 times total.
Classification:
| AMS MSC: | 30D20 (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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Pending Errata and Addenda
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