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Question

# Evaluate each of the following in $x + iy$ form, and compare with a computer solution. $\ln (-i)$

Solution

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$\textbf{\textcolor{#c34632}{Find :}}$ $w=\ln(-i)$

Let's assume $z=-i$ so the polar form of the number is

$z=r\exp(i\theta)\;\;\;\;\;\;\;\Rightarrow(1)$

The absolute $r$ is given by

$\boxed{r=|-i|=1}\;\;\;\;\;\;\;\Rightarrow(2)$

And the angle $\theta$ is given by

$\boxed{\theta=\dfrac{3\pi}{2}}$

Because the number locates in negative imaginary axis . so the number is

$\boxed{z=\exp(3\pi i/2)}$

So the the number $w$ is

\begin{align*} w=&\ln(r\exp(\theta i))\\ =&\ln(r)+(3\pi/2+2n\pi)i \;\;\;\;\;\;\;\;n=0,\pm 1 ,\pm 2 ,\pm 3 ,\dots\\ =&\ln(1) +(3\pi /2+2n\pi)i\;\;\;\;\;\;\;\;n=0,\pm 1 ,\pm 2 ,\pm 3 ,\dots\\ =&\boxed{(3\pi/2+2n\pi)i}\;\;\;\;\;\;\;\;n=0,\pm 1 ,\pm 2 ,\pm 3 ,\dots\\ \end{align*}

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