Let $f=u+iv$ be an entire function where $u,v$ are the real and imaginary parts of $f$ respectively. If the Jacobian $$ J_a= \left[ {\begin{array}{cc} u_x(a) & u_y(a) \\ v_x(a) & v_y(a) \\ \end{array} } \right] $$ is symmetric $\forall a\in \mathbb C$, then
(1) $f$ is a polynomial.
(2)$f$ is a polynomial of $\deg\le 1$.
(3)$f$ is necessarily a constant function.
(4)$f$ is a polynomial of degree strictly greater than 1.
$f$ is analytic. Using C-R equation, we have $\det J_a=u_x(a)v_y(a)-v_x(a)u_y(a)=u_x(a)u_x(a)+u_y(a)u_y(a)$. We know $J_a$ is symmetric $\implies u_y(a)=v_x(a)$. How do I proceed further. How to judge the option? Please help me.
The second Cauchy-Riemann equation says that the off-diagonal elements of the Jacobian differ by a sign, $v_x=-u_y$, so that to be symmetric, $v_x=u_y$, they have to be zero.
Which means that $u(x+iy)=p(x)$ is a function of $x$ alone and $v(x+iy)=q(y)$ of $y$ alone. Then the first CR-equation gives $p'(x)=u_x=v_y=q'(y)$ for all $x,y$ independent of each other which implies $$p'(x)=q'(0)=a=p'(0)=q'(y).$$ This gives $$f(z)=az+b$$ with real $a$ and complex $b$ as the only possible solutions and indeed these functions are all solutions.