本题目来源于试卷: Complex Numbers，类别为 IB数学
[问答题]
Consider $w=\frac{z-1}{z+\mathrm{i}} $ where $z=x+\mathrm{i}$ y and $\mathrm{i}=\sqrt{-1}$
1. If $z=\mathrm{i}$ ,
1. write w in the form $r \operatorname{cis} \theta$ ;
2. find the value of $w^{14}$ .
2. Show that in general,
$w=\frac{\left(x^{2}-x+y^{2}+y\right)+\mathrm{i}(y-x+1)}{x^{2}+(y+1)^{2}}$
3. Find condition under which $\operatorname{Re}(w)=1$ .
4. State condition under which w is:
1. real;
2. purely imaginary.
5. Find the modulus of z given that $\arg w=\frac{\pi}{4}$ .
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