I'm not 100% sure of the solution to the following problem:
Find all the solutions of:
, where z=x+iy
I attempted doing this question the way my lecture notes seem to indicate, as follows:
,
, where
But i'm really not sure this makes any sense?! Can anyone help me out of confusion? Is there another way of doing this or is this vaugely right?
That's a very interesting question. The answer is that there is no discrepancy between the two solutions, though it took me a while to figure out how they fit together.
Start with the solution in the form . There are two solutions for each value of n, namely
[1] and [2].
Looking at the second of these, the first thing you should notice is that 3–√10 is negative. In fact, (this follows from the fact that ). Therefore , and we can rewrite [2] as . This can be combined with [1] to form the single expression , which agrees with Plato's method for finding the solution.