GATE Mechanical 2026 | Question: 26

Question

If $w=\log _{e} z=\log _{e}(x+i y)$, where $i=\sqrt{-1}$, then which one of the following statements is correct?

• A. $w$ is analytic everywhere except at $z=0$
• B. $w$ is non-analytic everywhere
• C. The conjugate functions of $w$ are $\log _{e}\left(x^{2}+y^{2}\right)$ and $\log _{e}\left(x^{2}-y^{2}\right)$
• D. The conjugate functions of $w$ are $\tan ^{-1}(x / y)$ and $\tan ^{-1}(y / x)$