Question

The real and imaginary parts of $\log (1+i)=$

• A. $(\frac{1}{2},\frac{\pi }{4})$
• B. $(\log 2,\frac{\pi }{4})$
• C. $(\log \sqrt{2},\frac{\pi }{4})$
• D. $(\log \frac{1}{2},\frac{\pi }{4})$

Answer 1

The correct option is A $(\frac{1}{2},\frac{\pi }{4})$

$z=x+iy$ is the complex number and let $z=r{e}^{i\theta }$

$log\left(z\right)=log\left(r\right)+i\theta$

Where $r=\sqrt{x^2+y^2}$ and $\theta =ta{n}^{-1}y/x$

Here we have $z=log(1+i)$, $r=\sqrt{1+1}=\sqrt{2}$ and $\theta =ta{n}^{-1}1=\pi /4$

$log\left(\sqrt{2}\right)$ and $\pi /4$ are the real and imaginary parts of the $z$ respectively

option c gives the correct answer