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1. |z| < Argz.

Would this look like a spiral of increasing 'radius.' Like a swirly shape starting at the origin? Would the origin be included? I ask this because I don't think I can have |z| < 0.

Note: -pi < Argz <= pi.

2. log|z| = -2Argz.

Would I just exponentiate both sides to get [tex]\left| z \right| = e^{ - 2Argz} [/tex] ?

If that's correct then what would the shape look like? Perhaps a 'circle' with a a varying radius?

3. [tex]0 < Arg\left( {z - 1 - i} \right) < \frac{\pi }{3}[/tex]

I don't know how to work with this one. The part, z - 1 - i just means the difference between z and (1+i) I think. Let z = x + yi so [tex]0 < Arg\left( {\left( {x - 1} \right) + i\left( {y - 1} \right)} \right) < \frac{\pi }{3}[/tex].

Any help is appreciated.