GATE 2023 ME – Question 33

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

The value of $k$ that makes the complex-valued function $f(z) = e^{-kx}(\cos 2y - i\sin 2y)$ analytic, where $z = x + iy$, is _________.
(Answer in integer)

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Correct answer: 2

Explanation

Write $u = e^{-kx}\cos 2y$ and $v = -e^{-kx}\sin 2y$. The Cauchy-Riemann equations are $u_x = v_y$ and $u_y = -v_x$. Now $u_x = -ke^{-kx}\cos 2y$ and $v_y = -2e^{-kx}\cos 2y$, so $k = 2$. Also $u_y = -2e^{-kx}\sin 2y$ and $-v_x = -ke^{-kx}\sin 2y$, which agree for $k = 2$. So $k = 2$ (the function is then $e^{-2x}e^{-2iy} = e^{-2z}$).