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This is on my homework on differentials and partial differentiation, so I'm not sure what application these could have on the natural log of z

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Let $z=x+iy$. To find real and imaginary part of $\ln(x+iy)$ let $x+iy=re^{i\theta}$ where $r= \sqrt{x^2+y^2}$ and $\theta=\tan^{-1}{\frac{y}{x}}$ therefore, $$\ln(x+iy)=\ln\left(re^{i\theta}\right)=\ln(r)+i\theta \ln(e)= \ln(r)+i\theta$$ Hence, real part $$\ln(r)$$ i.e $$\ln\left(x^2+y^2\right)^{1/2}= \frac{1}{2}\ln\left(x^2+y^2\right)$$ and imaginary part is $$\theta=\tan^{-1}\frac{y}{x}$$

Notice here $$\ln(x+iy)= \ln(r)+ i\theta= \frac{1}{2}\ln\left(x^2+y^2\right)+i\tan^{-1}\frac{y}{x}$$

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