PlanetMath: periodicity of exponential function

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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 is always $\neq 0$ , we have

$\displaystyle e^\omega = 1.$

(1)

If we set $\omega := a+ib$ with

and

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

is real, that

. 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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one-periodic

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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 2828 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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