GATE 2024 EE – Question 30

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy integral formula, Taylor series, Laurent series, Residue theorem · 1 mark · Multiple select

Which of the following complex functions is/are analytic on the complex plane?

$f(z)=j\,\text{Re}(z)$
$f(z)=\text{Im}(z)$
$f(z)=e^{|z|}$
$f(z)=z^2-z$

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Correct answer: (D) $f(z)=z^2-z$

Explanation

A function is analytic only if it satisfies the Cauchy-Riemann equations. $j\,\text{Re}(z)$, $\text{Im}(z)$ and $e^{|z|}$ all fail them (they depend on $x,y$ but not through $z$ alone), while the polynomial $z^2-z$ is analytic everywhere.