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GATE 2014 EC (EC2) – Question 37

Engineering Mathematics · Complex Analysis · 2 marks · Multiple choice

The real part of analytic f(z), z = x + jy, is $e^{-y}\cos x$. Which is its imaginary part?

  1. $e^y\cos x$
  2. $e^{-y}\sin x$
  3. $-e^y\sin x$
  4. $-e^{-y}\sin x$

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Show answer and explanation

Correct answer: (B) $e^{-y}\sin x$

Explanation

For u=e⁻ʸcosx, the Cauchy-Riemann relations require vy=ux=−e⁻ʸsinx and vx=−uy=e⁻ʸcosx. Integration gives v=e⁻ʸsinx plus a real constant. The listed choice with zero additive constant is **e⁻ʸsinx,B**, consistent with f(z)=exp(jz).