Plus42 Equations, Preview Release

[Albert Chan]

I think complex powers with integer exponents should be done via squaring.
This is true even if result is not exact.

Y^X = |Y|^X * cis(arg(Y)*X)

Unlike pow() for real, we don't have excess precision to correct close to actual result.
Essentially, we are getting this, from Free42/Plus42:

Y^X ≈ round(round(|Y|)^X) * cis(round(round(arg(Y))*X))

Worst, argument inside cis() would required reduction, to within ±pi.

Python already does integer powers by squaring.
Complex pow (last column) not as good.

Python 3.9.5 (default, May 27 2021, 19:45:35)
[GCC 9.3.0] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> z = 3+4j
>>> q = 2+1j # sqrt(z)
>>> for i in range(10): print(i, q*z**i, z**(i+0.5))
...
0 (2+1j) (2+1j)
1 (2+11j) (2.0000000000000013+11j)
2 (-38+41j) (-37.999999999999986+41.00000000000001j)
3 (-278-29j) (-278-28.99999999999994j)
4 (-718-1199j) (-718.0000000000008-1198.9999999999995j)
5 (2642-6469j) (2641.9999999999964-6469.000000000001j)
6 (33802-8839j) (33801.99999999999-8839.000000000015j)
7 (136762+108691j) (136762.00000000006+108690.99999999996j)
8 (-24478+873121j) (-24477.99999999971+873121j)
9 (-3565918+2521451j) (-3565917.9999999995+2521451.000000001j)