3 ms·
I'm confused. How is this simpler? Is there something in (-1)^(2x) that can easily understood by staring at the complex plane? It seems mostly that you've gotte
by judofyr 1mo ago
I'm confused. How is this simpler? Is there something in (-1)^(2x) that can easily understood by staring at the complex plane? It seems mostly that you've gotten rid of "e", but one of the goals of Euler's formula IMO is to explain what "e^(i …)" means so I'm not sure how this variant is useful.
- voidmain 1mo agoI'll defend i^4x since I like it better. (cost x, sint x) is a point on the unit circle x turns counterclockwise from (1,0). cost x + i sint x is a point in the complex plane x turns counterclockwise from 1. Now look at integer powers of i, a point in the complex plane 1/4 turn from 1: i^0 = 1 (0 turns from 1) i^1 = i (1/4 turn from 1) i^2 = -1 (2/4 turn from 1) i^3 = -i (3/4 turn from 1) and we define complex exponentiation such that, for all real x, i^x = cost (x/4) + i sint (x/4) (x/4 turn from 1)