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asked 2012-11-02 06:42:21 -0500

MvG gravatar image

Complex argument of an algebraic number

This question is closely related to that question here. Basically I'd like to know whether there is a way to compute an accurate symbolic expression for the argument of an algebraic number.

That argument will in general not be an algebraic number itself, which seems to cause a lot of headache along the way. The following all fail, sometimes in rather spectacular backtracing ways:

sage: z = QQbar(3 + 2*I)
sage: z.arg()
AttributeError: 'AlgebraicNumber' object has no attribute 'arg'
sage: atan2(imag(z), real(z))
TypeError: Illegal initializer for algebraic number
sage: atan2(SR(imag(z)), SR(real(z)))
TypeError: Illegal initializer for algebraic number
sage: atan2(AA(imag(z)), AA(real(z)))
TypeError: Illegal initializer for algebraic number

I know a few cases which will work.

sage: atan2(QQ(imag(z)), QQ(real(z)))
arctan(2/3)

This however will break if the real or imaginary part were to contain any square roots.

sage: CC(z).arg()
0.588002603547568

This will give me a numeric approximation. I know I can get that approximation to arbitrary precision, but it's still not exact.

I have the impression that atan2 attempts to turn its result into an algebraic number, which will fail horribly. I would expect that result to contain an unevaluated call to atan2 instead, for the cases where the argument is not an algebraic number. Can this be done?