GATE 2024 EE Q.30: solution and answer

Official answer: DSource: Official answer key, GATE 2024 EE (IISc Bengaluru)

1. Write $z=x+jy$ and $f=u+jv$. If $f$ is differentiable at a point, the Cauchy-Riemann equations $u_x=v_y$ and $u_y=-v_x$ must hold there (a necessary condition). If the first partial derivatives are continuous and satisfy these equations, $f$ is differentiable (the sufficient condition). A function is analytic on the plane if it is differentiable at every point.

2. (A) $f=j\text{Re}(z)=jx$, so $u=0$ and $v=x$. The first equation holds: $u_x=0=v_y$. The second fails: $u_y=0$ but $-v_x=-1$. Cauchy-Riemann fails at every point, so (A) is not analytic.

3. (B) $f=\text{Im}(z)=y$, so $u=y$ and $v=0$. The first equation holds: $u_x=0=v_y$. The second fails: $u_y=1$ but $-v_x=0$. It fails at every point, so (B) is not analytic.

4. (C) $f=e^{|z|}=e^{\sqrt{x^2+y^2}}$ is real-valued, so $v=0$. Wherever $f$ is differentiable, Cauchy-Riemann would force $u_x=v_y=0$ and $u_y=-v_x=0$. But for $z\ne 0$, $u_x=\frac{x}{\sqrt{x^2+y^2}}{e}^{\sqrt{x^2+y^2}}\ne 0$ whenever $x\ne 0$. So Cauchy-Riemann fails at every point off the imaginary axis, and (C) is not analytic. (This is a general fact: a real-valued analytic function must be constant.)

5. (D) $f=z^2-z$ is a polynomial in $z$. Every polynomial is differentiable everywhere, here with $f^{\prime }(z)=2z-1$. As a check: $u=x^2-y^2-x$ and $v=2xy-y$ give $u_x=2x-1=v_y$ and $u_y=-2y=-v_x$ at every point, and these partial derivatives are continuous. So (D) is analytic on the whole plane (entire).

6. Only (D) is analytic.