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

$$ifz=x+iyfindlnzintermsofxandy$$$$$$

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

$$z=\mid z\mid e^{iarctan\left(\frac{im\left(z\right)}{re\left(z\right)}\right)}but\mid z\mid =\sqrt{x^2+y^2}\Rightarrow z=\sqrt{x^2+y^2}{e}^{iarctan\left(\frac{y}{x}\right)}$$$$ifx\ne 0\Rightarrow ln\left(z\right)=\frac{1}{2}ln(x^2+y^2)+iarctan\left(\frac{y}{x}\right)$$$$ifx=0\Rightarrow z=iy\Rightarrow ln\left(z\right)=ln\left(i\right)+ln\left(y\right)=\frac{i\pi }{2}+ln\left(y\right).$$

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

$$z=x+iy=\sqrt{x^2+y^2}{e}^{i{\tan }^{-1}\frac{y}{x}}$$$$\Rightarrow \mathbf{ln}\mathit{z}=\frac{\mathbf{ln}({\mathit{x}}^2+{\mathit{y}}^2)}{2}+\mathit{i}{\mathbf{tan}}^{-1}\frac{\mathit{y}}{\mathit{x}}$$

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