## 2016年11月9日水曜日

### 数学 - 解析学 - 数 - 複素数(複素数体の構成、実部・虚部・共役、絶対値)

$\begin{array}{l}1-3i+3{i}^{2}-{i}^{3}\\ =-2-2i\\ \\ \frac{\left(23+2i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}\\ =\frac{61}{25}+\frac{98}{25}i\\ \\ 1\left(n=4k\right)\\ i\left(n=4k+1\right)\\ -1\left(n=4k+2\right)\\ -i\left(n=4k+3\right)\end{array}$

$\begin{array}{l}|\frac{\alpha }{\beta }|=\sqrt{\frac{\alpha }{\beta }\overline{\left(\frac{\alpha }{\beta }\right)}}\\ =\sqrt{\frac{\alpha }{\beta }·\frac{\overline{\alpha }}{\overline{\beta }}}\\ =\sqrt{\frac{\alpha \overline{\alpha }}{\beta \overline{\beta }}}\\ =\frac{\sqrt{\alpha \overline{\alpha }}}{\sqrt{\beta \overline{\beta }}}\\ =\frac{|\alpha |}{|\beta |}\end{array}$

$\begin{array}{l}2·\sqrt{10}·\sqrt{20}\sqrt{2}=40\\ \frac{\sqrt{5}\sqrt{13}}{\sqrt{2}\sqrt{25}}=\frac{\sqrt{130}}{10}\end{array}$

$\begin{array}{l}{|\alpha +\beta |}^{2}+{|\alpha -\beta |}^{2}\\ =\left(\alpha +\beta \right)\overline{\left(\alpha +\beta \right)}+\left(\alpha -\beta \right)\overline{\left(\alpha -\beta \right)}\\ =\left(\alpha +\beta \right)\left(\overline{\alpha }+\overline{\beta }\right)+\left(\alpha -\beta \right)\left(\overline{\alpha }-\overline{\beta }\right)\\ ={|\alpha |}^{2}+{|\beta |}^{2}+\alpha \overline{\beta }+\overline{\alpha }\beta +{|\alpha |}^{2}+{|\beta |}^{2}-\alpha \overline{\beta }-\overline{\alpha }\beta \\ =2\left({|\alpha |}^{2}+{|\beta |}^{2}\right)\end{array}$

$\begin{array}{l}|\left(\alpha -\beta \right)+\beta |\le |\alpha -\beta |+|\beta |\\ ||\left(\alpha -\beta \right)+\beta ||-|\beta |\le |\alpha -\beta |\\ |\alpha |-|\beta |\le |\alpha -\beta |\\ |\alpha |-|\beta |\ge -|\alpha -\beta |\\ ||\alpha |-|\beta ||\le |\alpha -\beta |\end{array}$

$\begin{array}{l}|\left({\alpha }_{1}+{\alpha }_{2}+···+{\alpha }_{n}\right)+{\alpha }_{n+1}|\\ \le |{\alpha }_{1}+{\alpha }_{2}+···+{\alpha }_{n}|+|{\alpha }_{n+1}|\\ \le |{\alpha }_{1}|+|{\alpha }_{2}|+···+|{\alpha }_{n+1}|\end{array}$

$\begin{array}{l}|\frac{\alpha -\beta }{1-\overline{\alpha }\beta }|\\ =\frac{|\alpha -\beta |}{|1-\overline{\alpha }\beta |}\\ =\frac{\sqrt{\left(\alpha -\beta \right)\overline{\left(\alpha -\beta \right)}}}{\sqrt{\left(1-\overline{\alpha }\beta \right)\overline{\left(1-\overline{\alpha }\beta \right)}}}\\ =\frac{\sqrt{\left(\alpha -\beta \right)\left(\overline{\alpha }-\overline{\beta }\right)}}{\sqrt{\left(1-\overline{\alpha }\beta \right)\left(1-\alpha \overline{\beta }\right)}}\\ =\frac{\sqrt{{|\alpha |}^{2}-\alpha \overline{\beta }-\overline{\alpha }\beta +{|\beta |}^{2}}}{\sqrt{1-\alpha \overline{\beta }-\overline{\alpha }\beta +{|\alpha |}^{2}{|\beta |}^{2}}}\\ |\alpha |=1\\ \frac{\sqrt{1-\alpha \overline{\beta }-\overline{\alpha }\beta +{|\beta |}^{2}}}{\sqrt{1-\alpha \overline{\beta }-\overline{\alpha }\beta +{|\beta |}^{2}}}=1\\ |\beta |=1\\ \frac{\sqrt{{|\alpha |}^{2}-\alpha \overline{\beta }-\overline{\alpha }\beta +1}}{\sqrt{1-\alpha \overline{\beta }-\overline{\alpha }\beta +{|\alpha |}^{2}}}=1\end{array}$

$\begin{array}{l}1+{|\alpha |}^{2}{|\beta |}^{2}-\left({|\alpha |}^{2}+{|\beta |}^{2}\right)\\ =\left(1-{|\alpha |}^{2}\right)\left(1-{|\beta |}^{2}\right)>0\\ {|\alpha |}^{2}+{|\beta |}^{2}>{|\alpha |}^{2}+{|\beta |}^{2}\\ |\frac{\alpha -\beta }{1-\overline{\alpha }\beta }|<1\end{array}$

$\begin{array}{l}z=c+di\\ {z}^{2}={c}^{2}-{d}^{2}+2cdi\\ {c}^{2}-{d}^{2}=a\\ 2cd=b\\ {\left({c}^{2}+{d}^{2}\right)}^{2}={a}^{2}+{b}^{2}\\ {c}^{2}+{d}^{2}=\sqrt{{a}^{2}+{b}^{2}}\\ 2{c}^{2}=\sqrt{{a}^{2}+{b}^{2}}+a\\ {c}^{2}=\frac{a+\sqrt{{a}^{2}+{b}^{2}}}{2}\\ c=±\sqrt{\frac{a+\sqrt{{a}^{2}+{b}^{2}}}{2}}\\ {d}^{2}=\frac{-a+\sqrt{{a}^{2}+{b}^{2}}}{2}\\ d=±\sqrt{\frac{-a+\sqrt{{a}^{2}+{b}^{2}}}{2}}\\ z=±\left(\sqrt{\frac{a+\sqrt{{a}^{2}+{b}^{2}}}{2}}+\sqrt{\frac{a+\sqrt{{a}^{2}+{b}^{2}}}{2}}\right)\left(b>0\right)\\ z=±\left(\sqrt{\frac{a+\sqrt{{a}^{2}+{b}^{2}}}{2}}-\sqrt{\frac{a+\sqrt{{a}^{2}+{b}^{2}}}{2}}\right)\left(b<0\right)\end{array}$