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

This question checks whether you know where the complex logarithm fails to be analytic, and what its real and imaginary parts look like in Cartesian form.

Write \( z = x+iy = re^{i\theta} \), so \( w = \log_e z = \log_e r + i\theta = \dfrac{1}{2}\log_e(x^2+y^2) + i\tan^{-1}(y/x) \). The real part is \( u = \dfrac{1}{2}\log_e(x^2+y^2) \) and the imaginary part is \( v = \tan^{-1}(y/x) \).

1. \( w \) is analytic everywhere except at \( z=0 \): correct. The derivative \( dw/dz = 1/z \) exists everywhere \( z \neq 0 \). The only trouble point is the origin, where \( \log_e z \) is undefined.
2. \( w \) is non-analytic everywhere: wrong. \( u \) and \( v \) satisfy the Cauchy Riemann equations at every point away from the origin, so \( w \) is analytic there.
3. The conjugate functions are \( \log_e(x^2+y^2) \) and \( \log_e(x^2-y^2) \): wrong. The true real part carries a factor of one half, and the imaginary part is an inverse tangent, not another logarithm.
4. The conjugate functions are \( \tan^{-1}(x/y) \) and \( \tan^{-1}(y/x) \): wrong. Neither expression matches \( u \), and only \( \tan^{-1}(y/x) \) actually shows up, as the imaginary part alone.

So the correct statement is that \( w = \log_e z \) is analytic everywhere in the plane except at the single point \( z = 0 \).