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On a transcedental number

Prove that the number \arctan \left( \frac{1}{2} \right) is transcedental.

Solution

Let \arctan \left( \frac{1}{2} \right) = \alpha. Then

    \[e^{i \alpha} = \frac{2+i}{\sqrt{5}}\]

which is algebraic.  Hence it follows from Weierstrass – Lindemann theorem  that i \alpha , as algebraic, must be transcedental.

The exercise can also be found at mathematica.gr .

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