Find-the-value-of-ln-i-ln-3-4i-

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

Find the value of {\begin{cases} \ln i \\ \ln (3 + 4 i) \end{cases}

Answered by EDWIN88 last updated on 13/Feb/21

(1) \ln i = \ln ∣ i ∣ + i (\arg (i) + 2 n π) = \ln 1 + i (\frac{π}{2} + 2 n π)

(2) \ln (3 + 4 i) = \ln ∣ 3 + 4 i ∣= \ln \sqrt{3^{2} + 4^{2}} + i (\arctan (\frac{4}{3}) + 2 n π)

= \ln 5 + i (\arctan (\frac{4}{3}) + 2 n π)

principal value logarithms \log (z) set n = 0

{\begin{cases} \ln i = i \frac{π}{2} \\ \ln (3 + 4 i) = \ln 5 + i \arctan (\frac{4}{3}) \end{cases}

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