Can Arg( &#x03BD;<!-- ν --> ) be of the form

Brooke Ayala

Brooke Ayala

Answered question

2022-05-19

Can Arg( ν) be of the form π × μ for an irrational number μ?

Answer & Explanation

Brennen Bishop

Brennen Bishop

Beginner2022-05-20Added 6 answers

Yes. Try α + i β where α and β are rational and nonzero and arctan arctan ( β / α ) is not a rational multiple of π. The only cases where θ / π and tan ( θ ) are both rational are the "obvious" ones where tan ( θ ) { 0 , ± 1 }.

However, μ can't be an irrational algebraic number. This follows from the Gelfond-Schneider theorem, since ν / | ν | = e i μ π is a value of ( 1 ) μ .

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