Question Number 146923 by mathdanisur last updated on 16/Jul/21
$${if}\:\:\:\:{arg}\:\left(\frac{{i}\:-\:{z}}{{i}}\right)\:=\:\frac{\pi}{\mathrm{4}} \\ $$$${find}\:\:\:\:{RemZ}\:+\:{ImZ}\:=\:? \\ $$
Answered by gsk2684 last updated on 16/Jul/21
$${arg}\left(\mathrm{1}+{iz}\right)=\frac{\pi}{\mathrm{4}} \\ $$$${arg}\left(\mathrm{1}+{i}\left({x}+{iy}\right)\right)=\frac{\pi}{\mathrm{4}} \\ $$$${arg}\left(\left(\mathrm{1}−{y}\right)+{ix}\right)=\frac{\pi}{\mathrm{4}} \\ $$$$\mathrm{1}−{y}\:>\:\mathrm{0}\:,\:{x}>\mathrm{0},\mathrm{1}−{y}={x} \\ $$$${then}\:{Re}\:{z}\:+\:{Im}\:{z}\:=\:{x}+{y}=\mathrm{1}\: \\ $$$${where}\:\mathrm{1}>{y},\:{x}>\mathrm{0} \\ $$
Commented by SLVR last updated on 16/Jul/21
$${great}\:{sir} \\ $$
Commented by mathdanisur last updated on 16/Jul/21
$${thank}\:{you}\:{Ser} \\ $$