Which function is analytic in z = x + iy, with i=-1?

GATE 2025 PI – Question 37

Engineering Mathematics · Calculus · 2 marks · Multiple choice

Which function is analytic in z = x + iy, with $i=\sqrt{-1}$?

$e^x(\cos y+i\sin y)$
$e^x(\cos y-i\sin y)$
$e^x(-\cos y+i\sin y)$
$e^{-x}(\cos y+i\sin y)$

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Correct answer: (A) $e^x(\cos y+i\sin y)$

Explanation

Option A is $e^{x+iy}=e^z$, an entire analytic function. Its real and imaginary parts satisfy the Cauchy–Riemann equations everywhere. The alternatives depend nonanalytically on the conjugate variable. **A**.

GATE 2025 PI – Question 37

Explanation

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