Let f(z) be an analytic function, where z = x + iy. If the real part of f(z) is x y, and the imaginary part of f(z) is zero for y = 0, then f(z) is

GATE 2024 ME – Question 13

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

Let $f(z)$ be an analytic function, where $z = x + iy$. If the real part of $f(z)$ is $\cosh x\cos y$, and the imaginary part of $f(z)$ is zero for $y = 0$, then $f(z)$ is

$\cosh x\,\exp(-iy)$
$\cosh z\,\exp z$
$\cosh z\,\cos y$
$\cosh z$

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Correct answer: (D) $\cosh z$

Explanation

For $z = x + iy$, $\cosh z = \cosh x\cos y + i\sinh x\sin y$. Its real part is $\cosh x\cos y$ and its imaginary part $\sinh x\sin y$ is zero when $y = 0$, as required. It is analytic, and an analytic function is fixed by its real part up to an imaginary constant, which the condition on $y = 0$ sets to zero. So $f(z) = \cosh z$.