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GATE 2024 IN – Question 49

Engineering Mathematics · Complex variables · 2 marks · Multiple select

Both $f(z)=u(x,y)+iv(x,y)$ and $\overline{f(z)}=u(x,y)-iv(x,y)$ are analytic in a domain. Choose all correct statements.

  1. $\partial u/\partial x=\partial v/\partial y=0$
  2. $\partial u/\partial y=-\partial v/\partial x\ne0$
  3. $df(z)/dz=0$
  4. $df(z)/dz\ne0$

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Correct answer: (A) $\partial u/\partial x=\partial v/\partial y=0$; (C) $df(z)/dz=0$

Explanation

Cauchy–Riemann for $u+iv$ gives $u_x=v_y$ and $u_y=-v_x$. For $u-iv$, it gives $u_x=-v_y$ and $u_y=v_x$. Together all four partial derivatives vanish.

Hence the function is constant on the domain and $f^\prime(z)=0$. Correct choices: **A and C**.