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Question Number 132386 by liberty last updated on 13/Feb/21

Find the value of { ((ln i)),((ln (3+4i))) :}

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\begin{cases}{\mathrm{ln}\:{i}}\\{\mathrm{ln}\:\left(\mathrm{3}+\mathrm{4}{i}\right)}\end{cases} \\ $$

Answered by EDWIN88 last updated on 13/Feb/21

(1)ln i = ln ∣i∣ +i (arg (i)+2nπ)=ln 1+i ((π/2)+2nπ) (2)ln (3+4i)=ln ∣3+4i∣=ln (√(3^2 +4^2 )) +i(arctan ((4/3))+2nπ) = ln 5 +i(arctan ((4/3))+2nπ) principal value logarithms log (z) set n=0 { ((ln i = i (π/2))),((ln (3+4i) = ln 5 + i arctan ((4/3)))) :}

$$\left(\mathrm{1}\right)\mathrm{ln}\:{i}\:=\:\mathrm{ln}\:\mid{i}\mid\:+{i}\:\left(\mathrm{arg}\:\left({i}\right)+\mathrm{2}{n}\pi\right)=\mathrm{ln}\:\mathrm{1}+{i}\:\left(\frac{\pi}{\mathrm{2}}+\mathrm{2n}\pi\right) \\ $$$$\left(\mathrm{2}\right)\mathrm{ln}\:\left(\mathrm{3}+\mathrm{4}{i}\right)=\mathrm{ln}\:\mid\mathrm{3}+\mathrm{4}{i}\mid=\mathrm{ln}\:\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} }\:+{i}\left(\mathrm{arctan}\:\left(\frac{\mathrm{4}}{\mathrm{3}}\right)+\mathrm{2}{n}\pi\right) \\ $$$$\:=\:\mathrm{ln}\:\mathrm{5}\:+{i}\left(\mathrm{arctan}\:\left(\frac{\mathrm{4}}{\mathrm{3}}\right)+\mathrm{2}{n}\pi\right) \\ $$$$\mathrm{principal}\:\mathrm{value}\:\mathrm{logarithms}\:\mathrm{log}\:\left(\mathrm{z}\right)\:\mathrm{set}\:\mathrm{n}=\mathrm{0} \\ $$$$\:\begin{cases}{\mathrm{ln}\:{i}\:=\:{i}\:\frac{\pi}{\mathrm{2}}}\\{\mathrm{ln}\:\left(\mathrm{3}+\mathrm{4}{i}\right)\:=\:\mathrm{ln}\:\mathrm{5}\:+\:{i}\:\mathrm{arctan}\:\left(\frac{\mathrm{4}}{\mathrm{3}}\right)}\end{cases} \\ $$

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