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GATE 2026 ME – Question 36

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Multiple choice

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

  1. $w$ is analytic everywhere except at $z = 0$
  2. $w$ is non-analytic everywhere
  3. The conjugate functions of $w$ are $\log_e(x^2 + y^2)$ and $\log_e(x^2 - y^2)$
  4. The conjugate functions of $w$ are $\tan^{-1}(x/y)$ and $\tan^{-1}(y/x)$

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Show answer and explanation

Correct answer: (A) $w$ is analytic everywhere except at $z = 0$

Explanation

The logarithm is analytic at every point except the singular point $z = 0$ (ignoring the branch cut), because its real and imaginary parts, $\frac{1}{2}\log_e(x^2 + y^2)$ and $\tan^{-1}(y/x)$, satisfy the Cauchy-Riemann equations. The harmonic conjugate of the real part is $\tan^{-1}(y/x)$, so options C and D are wrong.