Locus question (1 Viewer)

[TuQuoi]

arg(z-i)-arg(z+i)>pi/4

I know how to sketch the locus if they're using an equals sign, but what happens when there's an inequality?

 

[TuQuoi]

Can you give a diagram of the inside part and the outside part? I can sort of visualise it, but I'm not sure whether the outside part includes the stuff after the line (if you make a line from point A to point B and to the left is the locus, the stuff on the right).

 

[Paradoxica]

For |z+1|>√2 AND Re(z)<0 the argument is less than π/4

For |z+1|=√2 AND Re(z)<0 the argument is exactly π/4

For |z+1|<√2 AND Re(z)<0 the argument is greater than π/4

The line Re(z)=0 is divided into two general cases and a degenerate case.

If |Im(z)| > 1, then the argument is ambiguous (0 or 2π) and the inequality is not defined.

If |Im(z)| < 1, then the argument is π.

If |Im(z)| = 1, then you have Arg(0) which is undefined.

The argument for Re(z)>0 is well defined and satisfies the inequality.