${\displaystyle ({\sqrt {3}}+\mathrm {i} )^{n}=2^{n}\mathrm {e} ^{\mathrm {i} n\pi /6}}$ est :

1°  réel positif si ${\displaystyle {\frac {n\pi }{6}}\equiv 0\mod {2\pi }}$, c'est-à-dire ${\displaystyle 12\mid n}$ ;

2°  imaginaire pur si ${\displaystyle {\frac {n\pi }{6}}\equiv {\frac {\pi }{2}}\mod {\pi }}$, c'est-à-dire ${\displaystyle n\equiv 3\mod 6}$.

${\displaystyle z_{1}=z^{3}z_{2}}$ donc

${\displaystyle {\begin{aligned}z_{1},z_{2}\in \mathbb {R} &\Leftrightarrow z_{2},z^{3}\in \mathbb {R} \\&\Leftrightarrow z\left(z+1\right),z^{3}\in \mathbb {R} \\&\Leftrightarrow z^{2}+z+1,z^{3}-1\in \mathbb {R} \\&\Leftrightarrow z^{2}+z+1,\left(z^{2}+z+1\right)\left(z-1\right)\in \mathbb {R} \\&\Leftrightarrow z^{2}+z+1=0\end{aligned}}}$

(car par hypothèse, ${\displaystyle z-1\notin \mathbb {R} }$).

Les solutions ${\displaystyle z}$ sont donc ${\displaystyle \operatorname {e} ^{\mathrm {i} 2\pi /3}}$ et ${\displaystyle \operatorname {e} ^{-\mathrm {i} 2\pi /3}}$, et dans les deux cas, ${\displaystyle z_{1}=z_{2}=-1}$.

Soit ${\displaystyle \lambda :={\frac {z'}{z}}}$.

1. ${\displaystyle |z+z'|=|z|+|z'|\Leftrightarrow |1+\lambda |=1+|\lambda |\Leftrightarrow |1+\lambda |^{2}=\left(1+|\lambda |\right)^{2}\Leftrightarrow \left(1+\lambda \right)\left(1+{\overline {\lambda }}\right)=1+2|\lambda |+|\lambda |^{2}\Leftrightarrow \operatorname {Re} (\lambda )=|\lambda |\Leftrightarrow \lambda \in \mathbb {R} _{+}}$.
2. De même, ${\displaystyle |z+z'|=|z|-|z'|\Leftrightarrow |1-\lambda |=1-|\lambda |\Leftrightarrow \left[|\lambda |\leq 1{\text{ et }}|1-\lambda |^{2}=\left(1-|\lambda |\right)^{2}\right]\Leftrightarrow \left[|\lambda |\leq 1{\text{ et }}\operatorname {Re} (\lambda )=-|\lambda |\right]\Leftrightarrow \lambda \in \left[-1,0\right]}$.

Soit ${\displaystyle u:=z+4\mathrm {i} }$. Ainsi, ${\displaystyle Z={\frac {u}{\bar {u}}}}$.

1°  ${\displaystyle Z}$ est défini si et seulement si ${\displaystyle u\neq 0}$ c'est-à-dire ${\displaystyle z\neq -4\mathrm {i} }$, et dans ce cas, ${\displaystyle |Z|=1}$.

2°

1. ${\displaystyle Z={\overline {Z}}\Leftrightarrow {\frac {u}{\bar {u}}}={\frac {\bar {u}}{u}}\Leftrightarrow u^{2}={\overline {u^{2}}}\Leftrightarrow u^{2}\in \mathbb {R} \Leftrightarrow u\in \mathbb {R} \cup \mathrm {i} \mathbb {R} \Leftrightarrow z\in \left(\mathbb {R} -4\mathrm {i} \right)\cup \mathrm {i} \mathbb {R} }$.
2. De même, ${\displaystyle Z=-{\overline {Z}}\Leftrightarrow u^{2}\in \mathrm {i} \mathbb {R} \Leftrightarrow u\in \operatorname {e} ^{\mathrm {i} \pi /4}\mathbb {R} \cup \operatorname {e} ^{-\mathrm {i} \pi /4}\mathbb {R} \Leftrightarrow z\in \left(\operatorname {e} ^{\mathrm {i} \pi /4}\mathbb {R} -4\mathrm {i} \right)\cup \left(\operatorname {e} ^{-\mathrm {i} \pi /4}\mathbb {R} -4\mathrm {i} \right)}$.

${\displaystyle \forall x,y\in \mathbb {R} \quad f\left(x+\mathrm {i} y\right)=x^{2}+y^{2}-x+y+\mathrm {i} \left(y-x+1\right)}$.

1. ${\displaystyle f\left(x+\mathrm {i} y\right)\in \mathbb {R} \Leftrightarrow y=x-1}$ (droite).
2. ${\displaystyle f\left(x+\mathrm {i} y\right)\in \mathrm {i} \mathbb {R} \Leftrightarrow \left(x-{\frac {1}{2}}\right)^{2}+\left(y+{\frac {1}{2}}\right)^{2}={\frac {1}{2}}}$ (cercle).

Soient ${\displaystyle \left(x,y\right)}$ les coordonnées de M.

1. ${\displaystyle f(z)\in \mathbb {R} \Leftrightarrow {\frac {z+1}{{\bar {z}}-1}}={\frac {{\bar {z}}+1}{z-1}}\Leftrightarrow z^{2}-1={\bar {z}}^{2}-1\Leftrightarrow z^{2}\in \mathbb {R} \Leftrightarrow xy=0}$ (union des deux axes).
2. ${\displaystyle f(z)\in \mathrm {i} \mathbb {R} \Leftrightarrow {\frac {z+1}{{\bar {z}}-1}}=-{\frac {{\bar {z}}+1}{z-1}}\Leftrightarrow z^{2}-1=1-{\bar {z}}^{2}\Leftrightarrow z^{2}+{\bar {z}}^{2}=2\Leftrightarrow \operatorname {Re} \left(z^{2}\right)=1\Leftrightarrow x^{2}-y^{2}=1}$ (hyperbole).

1°  Soit ${\displaystyle z:=1+\mathrm {i} }$.

${\displaystyle (1+\mathrm {i} )^{n}+(1-\mathrm {i} )^{n}=z^{n}+{\bar {z}}^{n}=z^{n}+{\overline {z^{n}}}=2\operatorname {Re} \left(z^{n}\right)}$ ;

${\displaystyle (1+\mathrm {i} )^{n}-(1-\mathrm {i} )^{n}=z^{n}-{\bar {z}}^{n}=z^{n}-{\overline {z^{n}}}=2\mathrm {i} \operatorname {Im} \left(z^{n}\right)}$.

2°  Soient ${\displaystyle a:={\binom {n}{0}}-{\binom {n}{2}}+{\binom {n}{4}}-{\binom {n}{6}}+\cdots }$ et ${\displaystyle b:={\binom {n}{1}}-{\binom {n}{3}}+{\binom {n}{5}}-{\binom {n}{7}}+\cdots }$.

${\displaystyle {\begin{aligned}a+\mathrm {i} b&=\sum _{k=0}^{n}{\binom {n}{k}}\mathrm {i} ^{k}=(1+\mathrm {i} )^{n}=\left({\sqrt {2}}\operatorname {e} ^{\mathrm {i} \pi /4}\right)^{n}=2^{n/2}\operatorname {e} ^{n\mathrm {i} \pi /4}\\&={\begin{cases}(-4)^{m}&{\text{si }}n=4m\\(-4)^{m}(1+\mathrm {i} )&{\text{si }}n=4m+1\\(-4)^{m+1}(-\mathrm {i} )&{\text{si }}n=4m+2\\(-4)^{m+1}(1-\mathrm {i} )&{\text{si }}n=4m+3.\end{cases}}\end{aligned}}}$