Implement Pow for Complex

[clarfonthey]

See rust-num/num-traits#38 for details on this trait.

[shingtaklam1324]

There is the issue of raising a Complex<T> to a power of Complex<T>, as it is equal to

a^(b+ic) = (a ^ b)[cos(c ln a) + i sin(c ln a)]

If a is a complex number, then ln ahas multiple answers, and there is not one definite answer. Instead, there is an infinite set of possible answers.

[clarfonthey]

@shingtaklam1324 right, but returning the principal solution is usually what people want. It's like how asin(1)/π is really (8n + 1)/4 but any practical implementation will return 1/4 as you can get any other solution by adding 2n.

Complex numbers work similarly; solutions can be obtained by repeatedly multiplying a constant complex number, which I don't recall the equation for off the top of my head. The point is that the principal solution is 99% of the time what you want.

I mean, in theory you could return an iterator over the solutions because they're countable, but I don't know if that's appropriate outside of niche cases.

[cuviper]

FWIW, we do already have powc, and rust-num/num#137 (comment) referenced this derivation.

[shingtaklam1324]

@clarcharr I think I may have over complicated this slightly, in that case, it should just be a simple refactoring or having a wrapper function for powc to

impl Pow<Complex<T>> for Complex<T>
  where T: Float + Clone

and powf to

impl Pow<T> for Complex<T>
  where T: Float + Clone

[cuviper]

Yes, I would just wrap powc and powf. It might be nice to add powi too.

[clarfonthey]

Floats currently support ints and floats as exponents so I'd do all three for Complex.

[benruijl]

powi would be very welcome! I get some rounding errors when using powf:

        println!(
            "test: {:?}",
            Complex::new(0., -2. * f64::PI()).powf(1.)
        );

Yields: Complex { re: 0.00000000000000038473413874435795, im: -6.283185307179586 }