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GATE 2016 ME (ME2) – Question 14

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 1 mark · Numerical answer

For $f(z)=u(x,y)+iv(x,y)$ with $z=x+iy$, $u=2kxy$ and $v=x^2-y^2$, the value of k for analyticity is ______.

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Correct answer: -1.1 to -0.9

Explanation

Cauchy–Riemann equations require ux=vy and uy=−vx. For u=2kxy,v=x²−y², these give2ky=−2y and2kx=−2x throughout a region. Both require **k=−1**. Checking only at the origin would not establish analyticity in a neighbourhood.