Mathematik-Online lexicon: Irrational and Imaginary Exponents

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	Mathematik-Online lexicon:
	Irrational and Imaginary Exponents

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For irrational and purely imaginary exponents, one generally obtains infinitely many solutions, as shown in the following examples:

infinitely many solutions on the unit circle:

$\displaystyle \mathrm{i}^{\pi}$ $\displaystyle =$ $\displaystyle \exp ((\pi/2+2\pi k)\mathrm{i})^{\pi}$

$\displaystyle =$ $\displaystyle \exp \left(\mathrm{i}[\pi^2/2+2\pi^2 k]\right),\qquad k \in \mathbb{Z}$
infinitely many solutions on a half line:

$\displaystyle \pi^{\mathrm{i}}$ $\displaystyle =$ $\displaystyle \exp (\ln \pi + 2\pi k\mathrm{i})^{\mathrm{i}} = \exp (\mathrm{i} \ln \pi -2\pi k)$

$\displaystyle =$ $\displaystyle \exp(-2\pi k) \exp(\mathrm{i}\ln \pi), \qquad k \in \mathbb{Z}$
infinitely many solutions on the positive real line:

$\displaystyle \mathrm{i}^{\mathrm{i}}$ $\displaystyle =$ $\displaystyle \exp ((\pi/2+2\pi k)\mathrm{i})^{\mathrm{i}}$

$\displaystyle =$ $\displaystyle \exp (-\pi/2-2\pi k), \qquad k \in \mathbb{Z}$

see also:

automatisch erstellt am 15. 11. 2013

$\displaystyle \mathrm{i}^{\pi}$	$\displaystyle =$	$\displaystyle \exp ((\pi/2+2\pi k)\mathrm{i})^{\pi}$
	$\displaystyle =$	$\displaystyle \exp \left(\mathrm{i}[\pi^2/2+2\pi^2 k]\right),\qquad k \in \mathbb{Z}$

$\displaystyle \pi^{\mathrm{i}}$	$\displaystyle =$	$\displaystyle \exp (\ln \pi + 2\pi k\mathrm{i})^{\mathrm{i}} = \exp (\mathrm{i} \ln \pi -2\pi k)$
	$\displaystyle =$	$\displaystyle \exp(-2\pi k) \exp(\mathrm{i}\ln \pi), \qquad k \in \mathbb{Z}$

$\displaystyle \mathrm{i}^{\mathrm{i}}$	$\displaystyle =$	$\displaystyle \exp ((\pi/2+2\pi k)\mathrm{i})^{\mathrm{i}}$
	$\displaystyle =$	$\displaystyle \exp (-\pi/2-2\pi k), \qquad k \in \mathbb{Z}$