Consider the circle \(|z-1|=1\).

a. Rewrite the equation in rectangular coordinates by setting \(z=\) \(x+i y\).

b. Sketch the resulting circle using part a.

c. Consider the image of the circle under the mapping \(f(z)=z^{2}\), given by \(\left|z^{2}-1\right|=1\).

i. By inserting \(z=r e^{i \theta}=r(\cos \theta+i \sin \theta)\), find the equation of the image curve in polar coordinates.

ii. Sketch the image curve. You may need to refer to your Calculus II text for polar plots. [Maple might help.]