Large powers of complex numbers

[bjnartowt]

Homework Statement

Suppose you raise a complex number to a very large power, z^n, where z = a + ib, and n ~ 50, 500, one million, etc. On raising to such a large power, the argument will shift by n*ArcTan[b/a] mod 2*Pi, and this is easy to see analytically. However, is there less numerical error when z remains in rectangular form, or less when it is converted to rectangular form?

Homework Equations

The Attempt at a Solution

 

[voko]

What do you think?

 

[bjnartowt]

Honestly, I think I'll have to compute Z analytically, and not leave it to a machine to do it.

 

[voko]

The problem is about where the error is greater, assuming either method is done numerically.