Question

If \( w = \log_e z = \log_e(x+iy) \), 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 \( \log_e(x^2+y^2) \) and \( \log_e(x^2-y^2) \)
• D. The conjugate functions of \( w \) are \( \tan^{-1}(x/y) \) and \( \tan^{-1}(y/x) \)

Hint:

Split w into its real and imaginary parts using z = re^{i theta} and check where the Cauchy-Riemann equations fail.

Answer 1

The Correct Option is A

A cleaner way to settle this is to test the Cauchy Riemann equations directly on $u$ and $v$ instead of quoting the known derivative of $\log z$.

With $u = \frac{1}{2}\log_e(x^2+y^2)$ and $v = \tan^{-1}(y/x)$, take partial derivatives: $u_x = \frac{x}{x^2+y^2}$, $u_y = \frac{y}{x^2+y^2}$, $v_x = \frac{-y}{x^2+y^2}$, $v_y = \frac{x}{x^2+y^2}$.

Check $u_x = v_y$: both equal $\frac{x}{x^2+y^2}$, true. Check $u_y = -v_x$: $u_y = \frac{y}{x^2+y^2}$ and $-v_x = \frac{y}{x^2+y^2}$, also true. Both hold at every point except $x^2+y^2=0$, meaning $z=0$, so $w$ is analytic on the whole plane except that point.

1. Option A matches this result exactly, so it is the right choice.
2. Option B fails because the Cauchy Riemann equations hold almost everywhere, not nowhere.
3. Option C drops the factor of one half in $u$ and swaps a tangent for a second logarithm in $v$, both wrong.
4. Option D mixes up $x/y$ and $y/x$ and calls both a conjugate function, when only one of them is actually $v$.

The Cauchy Riemann check confirms option A: analytic everywhere except at $z = 0$.