Regneregler

1.1.13.2 Regneregler

$\mathit{A}\mathrm{\pm }\mathit{B}\mathrm{=}\mathrm{(}{\mathit{a}}_{\mathrm{1}}\mathrm{\pm }{\mathit{b}}_{\mathrm{1}}\mathrm{)}\mathrm{+}\mathit{i}\mathrm{(}{\mathit{a}}_{\mathrm{2}}\mathrm{\pm }{\mathit{b}}_{\mathrm{2}}\mathrm{)}$ [1.1.226]

$\mathit{A}\mathrm{\cdot }\mathit{B}\mathrm{=}\mathrm{(}{\mathit{a}}_{\mathrm{1}}{\mathit{b}}_{\mathrm{1}}\mathrm{–}{\mathit{a}}_{\mathrm{2}}{\mathit{b}}_{\mathrm{2}}\mathrm{)}\mathrm{+}\mathit{i}\mathrm{(}{\mathit{a}}_{\mathrm{1}}{\mathit{b}}_{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}{\mathit{b}}_{\mathrm{1}}\mathrm{)}$ [1.1.227]

$\frac{\mathit{B}}{\mathit{A}}\mathrm{=}\frac{{\mathit{b}}_{\mathrm{1}}\mathrm{+}\mathit{i}{\mathit{b}}_{\mathrm{2}}}{{\mathit{a}}_{\mathrm{1}}\mathrm{+}\mathit{i}{\mathit{a}}_{\mathrm{2}}}\mathrm{=}\frac{{\mathit{a}}_{\mathrm{1}}{\mathit{b}}_{\mathrm{1}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}{\mathit{b}}_{\mathrm{2}}}{{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}}\mathrm{+}\mathit{i}\frac{{\mathit{a}}_{\mathrm{1}}{\mathit{b}}_{\mathrm{2}}\mathrm{–}{\mathit{a}}_{\mathrm{2}}{\mathit{b}}_{\mathrm{1}}}{{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}}$ [1.1.228]

$\frac{\mathrm{1}}{\mathit{A}}\mathrm{=}\frac{{\mathit{a}}_{\mathrm{1}}}{{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}}\mathrm{–}\mathit{i}\frac{{\mathit{a}}_{\mathrm{2}}}{{\mathit{a}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathrm{2}}^{\mathrm{2}}}$ [1.1.229]

$\frac{\mathrm{1}}{\mathit{i}}\mathrm{=}\mathrm{–}\mathit{i}$ [1.1.230]

$\mathrm{\left|}\mathit{A}\mathit{B}\mathrm{\right|}\mathrm{=}\mathrm{\left|}\mathit{A}\mathrm{\right|}\mathrm{\cdot }\mathrm{\left|}\mathit{B}\mathrm{\right|}$ [1.1.231]

$\mathrm{\left|}\mathrm{\right|}\mathit{A}\mathrm{|}\mathrm{–}\mathrm{|}\mathit{B}\mathrm{\left|}\mathrm{\right|}\mathrm{\le }\mathrm{|}\mathit{A}\mathrm{\pm }\mathit{B}\mathrm{|}\mathrm{\le }\mathrm{\left|}\mathit{A}\mathrm{\right|}\mathrm{+}\mathrm{\left|}\mathit{B}\mathrm{\right|}$ [1.1.232]

$\mathrm{\left|}\frac{\mathit{B}}{\mathit{A}}\mathrm{\right|}\mathrm{=}\frac{\mathrm{\left|}\mathit{B}\mathrm{\right|}}{\mathrm{\left|}\mathit{A}\mathrm{\right|}}$ [1.1.233]

$\mathit{A}\mathrm{=}\mathrm{\left|}\mathit{A}\mathrm{\right|}\mathrm{(}\cos \arg \mathit{A}\mathrm{+}\mathit{i}\sin \arg \mathit{A}\mathrm{)}$ [1.1.234]

$\arg \mathrm{(}\mathit{A}\mathrm{\cdot }\mathit{B}\mathrm{)}\mathrm{=}\arg \mathit{A}\mathrm{+}\arg \mathit{B}$[1.1.235]

$\arg \frac{\mathit{A}}{\mathit{B}}\mathrm{=}\arg \mathit{A}\mathrm{–}\arg \mathit{B}$ [1.1.236]

${\mathit{A}}^{\mathrm{*}}\mathrm{\pm }{\mathit{B}}^{\mathrm{*}}\mathrm{=}\mathrm{(}\mathit{A}\mathrm{\pm }\mathit{B}{\mathrm{)}}^{\mathrm{*}}$ [1.1.237]

${\mathit{A}}^{\mathrm{*}}\mathrm{\cdot }{\mathit{B}}^{\mathrm{*}}\mathrm{=}\mathrm{(}\mathit{A}\mathit{B}{\mathrm{)}}^{\mathrm{*}}$ [1.1.238]

$\frac{{\mathit{B}}^{\mathrm{*}}}{{\mathit{A}}^{\mathrm{*}}}\mathrm{=}\mathrm{(}\frac{\mathit{B}}{\mathit{A}}{\mathrm{)}}^{\mathrm{*}}$ [1.1.239]

${\mathit{A}}^{\mathrm{*}}\mathit{A}\mathrm{=}\mathrm{|}\mathit{A}{\mathrm{|}}^{\mathrm{*}}$ [1.1.240]

Eulers formler

$\cos \mathit{x}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\cdot }\mathrm{(}{\mathit{e}}^{\mathit{i}\mathit{x}}\mathrm{+}{\mathit{e}}^{\mathrm{–}\mathit{i}\mathit{x}}\mathrm{)}$ [1.1.241]

$\sin \mathit{x}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\cdot }\mathrm{(}{\mathit{e}}^{\mathit{i}\mathit{x}}\mathrm{–}{\mathit{e}}^{\mathrm{–}\mathit{i}\mathit{x}}\mathrm{)}$ [1.1.242]

${\mathit{e}}^{\mathit{i}\mathit{x}}\mathrm{=}\cos \mathit{x}\mathrm{+}\mathit{i}\sin \mathit{x}$ [1.1.243]

${\mathit{e}}^{\mathrm{–}\mathit{i}\mathit{x}}\mathrm{=}\cos \mathit{x}\mathrm{–}\mathit{i}\sin \mathit{x}$ [1.1.244]

Moivres formel

$\cos \mathit{n}\mathit{x}\mathrm{+}\mathit{i}\sin$ $\mathit{n}\mathit{x}\mathrm{=}\mathrm{(}\cos \mathit{x}\mathrm{+}\mathit{i}\sin \mathit{x}{\mathrm{)}}^{\mathit{n}}$ [1.1.245]