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GATE 2023 IN – Question 39

Engineering Mathematics · Complex variables · 2 marks · Multiple choice

Let $f(z)=j(1-z)/(1+z)$, where $z$ is complex and $j=\sqrt{-1}$. Its inverse $f^{-1}$ maps the real axis to the ______.

  1. unit circle with centre at the origin
  2. unit circle with centre not at the origin
  3. imaginary axis
  4. real axis

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Show answer and explanation

Correct answer: (A) unit circle with centre at the origin

Explanation

Solve $w=j(1-z)/(1+z)$ to get $z=(j-w)/(j+w)$. For real $w$, both magnitudes equal $\sqrt{1+w^2}$, hence $|z|=1$. The real axis maps to the unit circle centred at the origin, **A**, with one limiting point corresponding to infinity.