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Transcedental roots

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Tolaso J Kos
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Transcedental roots

#1

Post by Tolaso J Kos » Sat Nov 21, 2015 12:31 am

Prove that the non zero roots of the equation $\tan x =x $ are transcedental.
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Re: Transcedental roots

#2

Post by Riemann » Sat Oct 22, 2016 10:06 am

We have that:

$$\tan x = \frac{e^{ix}- e^{-ix}}{i \left ( e^{ix} +e^{-ix} \right )}= -i \frac{e^{2ix}- 1}{e^{2ix}+1} \Rightarrow e^{2ix} = \frac{1+i\tan x}{1-i \tan x}$$

If $x$ is an algebraic number, then $2ix$ is also algebraic hence from Lindemann's theorem $e^{2ix}$ is transcedental (for $x\neq 0$). Hence $\tan x$ must also be transcedental.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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