Find the real constants $a,b,c,d$ so that the function is analytic. $f(z)=x^2+axy+by^2+ i(cx^2+dxy+y^2)$
I know that,since the given function is analytic,we can use cauchy reimann equations to solve this, I got :
$$\frac{\partial u}{\partial x} = 2x + ay \\$$
$$\frac{\partial v}{\partial y} = 2y + xd \\$$
$$\frac{\partial u}{\partial y} = ax + 2by \\$$
$$\frac{\partial v}{\partial x} = 2cx + yd$$
I know that we have to equate this but even though I equate I get an equation not a solution for the constants. At last I got $a=d$ and $c=d$ I don't know whether this is right or wrong, could you please help me?
You must have $u_x=v_y$ and $u_y=-v_x$, hence: $$\begin{cases}2x+ay=2y+dx \\ ax+2by=-2cx-dy\end{cases}$$ From the first equation you get $a=d=2$, can you take it from here?