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GATE 2020 ME (ME1) – Question 46

Engineering Mathematics · Complex Variables: Analytic functions, Cauchy-Riemann equations, integral theorem, Taylor and Laurent series · 2 marks · Numerical answer

An analytic function of $z=x+iy$, $i=\sqrt{-1}$, is $f(z)=x^2-y^2+i\phi(x,y)$, where φ is real. Find the imaginary part of f at z = 1 + i, to two decimals.

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Correct answer: 1.99 to 2.01

Explanation

Cauchy–Riemann equations give ψy=2x and ψx=2y, hence ψ=2xy+C. Under the intended zero imaginary constant, ψ(1,1)=**2.00**. Strictly, the stated real part alone does not determine C; a normalization such as f(0)=0 is needed for uniqueness.