if-z-x-iy-find-lnz-interms-of-x-and-y-

Question Number 67518 by mathmax by abdo last updated on 28/Aug/19

i f z = x + i y f i n d l n z i n t e r m s o f x a n d y

Commented by mathmax by abdo last updated on 30/Aug/19

z =∣ z ∣ e^{i a r c t a n (\frac{i m (z)}{r e (z)})} b u t ∣ z ∣= \sqrt{x^{2} + y^{2}} \Rightarrow z = \sqrt{x^{2} + y^{2}} e^{i a r c t a n (\frac{y}{x})}

i f x \neq 0 \Rightarrow l n (z) = \frac{1}{2} l n (x^{2} + y^{2}) + i a r c t a n (\frac{y}{x})

i f x = 0 \Rightarrow z = i y \Rightarrow l n (z) = l n (i) + l n (y) = \frac{i π}{2} + l n (y) .

Answered by mr W last updated on 28/Aug/19

z = x + i y = \sqrt{x^{2} + y^{2}} e^{i \tan^{- 1} \frac{y}{x}}

\Rightarrow \ln z = \frac{\ln (x^{2} + y^{2})}{2} + i \tan^{- 1} \frac{y}{x}

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