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  • If |z1| = |z2| = ….. = |zn| = 1, then

 If |z_1| = |z_2| =... = |z_n| = 1, then

Show that |z_1 + z_2 + z_3 +... + z_n| =|1/z_1 +1/z_2 +...+1/z_n |

 

Answers (1)

|z_1| = |z_2| =... = |z_n| = 1

|z_1|^2 = |z_2|^2 =... = |z_n|^2 = 1

z_1 \bar{z_1 } =z_2 \bar{z_2 }=...=z_n \bar{z_n }

z_1=\frac{1}{\bar{z_1}},z_2=\frac{1}{\bar{z_2}},...z_n=\frac{1}{\bar{z_n}}

Now, |z_1+z_2+z_3+...z_n | =\left | \frac{z_1\bar{z_1}}{\bar{z_1}}+\frac{z_2\bar{z_2}}{\bar{z_2}}+...+\frac{z_n\bar{z_n}}{\bar{z_n}} \right |

=\left | \frac{1}{\bar{z_1}}+\frac{1}{\bar{z_2}}+...+\frac{1}{\bar{z_n}} \right | 

=\left | {\frac{1}{z_1}+\frac{1}{z_2}+...+\frac{1}{z_n}} \right |

Posted by

infoexpert21

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