I need the real and imaginary part of $\log \sin (x+iy)$. I expand $\sin(x+iy)=\sin x \cosh y+i \cos x \sinh y$. But I don't know how to do it s logarithm
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See https://en.wikipedia.org/wiki/Complex_logarithm – user121049 Sep 23 '17 at 07:25
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1For $f=u+iv$ we have $f=|f|e^{i\phi}$ and $\log f=\ln|f|+i(\phi+2\pi k)$. Note that $|f|=\sqrt{u^2+v^2}$ and $\phi=\mathrm{atan2}(v,u)$ – Fakemistake Sep 23 '17 at 07:29
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$$\sin(x+iy) = \sin x \cosh y + i \cos x \sinh y$$
$$\sin(x+iy) = \sqrt{\sin^2 x \cosh^2 y + \cos^2 x \sinh^2 y} \exp(i \arctan(\cot x \tanh y))$$
$$\ln \sin(x+iy) = \dfrac 1 2 \ln (\sin^2 x \cosh^2 y + \cos^2 x \sinh^2 y) + i \arctan(\cot x \tanh y)$$
Kenny Lau
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