Find a real-valued function which satisfied Cauchy-Riemann equations

Question

Question: Find a real-valued function $v(x,y)$ such that $v(0,0)=1$ and together, $u$ and $v$ satisfy the Cauchy-Riemann equations in the entire complex plane.

Let $v(x,y)=M(x,y)+i.0$ which is a real-valued function.

CR equations for $v(x,y)$,

$$M_x=0$$

$$M_y=0$$

But it seems I am going to a wrong direction. Any solution or hints will be great helpful for me.

Answer 1

You can take $u(x,y)=x$ and $v(x,y)=y+1$. Then, $u_x=v_y=1$ and $u_y=-v_x=0$. Besides, $v(0,0)=1$.