Suppose that $u(x,y) + i v(x,y) = x^{3} − kxy^{2} + 12xy − 12x$ for some constant $k\in Complex$ $plane$.Find all values of $k$ for which $u$ is the real part of a holomorphic function.
I know that I should use the Cauchy-Riemann equations , but in the above given function I'm not sure what to set v equal to. Should it be zero or am I missing something ?
Best Answer
Using common notation your $x$ and $y$ live inside the real numbers. So it would seem that, for $k=a+bi$, you could set $v=-bxy^2 $ and $u=x^3-axy^2+12xy-12x$. You could then use Cauchy-Riemann.