This question was asked in my complex analysis quiz and I am getting wrong answer. So, I am looking for the right argument in this question.
Let f be a real valued harmonic function on $\mathbb{C}$ . Define the functions
g= $\frac{\partial f}{\partial x}$ -i $\frac{\partial f}{\partial y}$, $h=\frac{\partial f}{\partial x}$ +i $\frac{\partial f}{\partial y}$
Then which of $g$ and $h$ is holomorphic function?
Answer:
g is holomorphic , but f is not
I thought of using the result that if f is a harmonic function then there exists an holomorphic function whose real part is f and a holomorphic function whose imagenary part is f but I can't be sure which is that holomorphic function so it cant be used. I can't think of any other result other than this.
Background:I have studied about harmonic function sfrom ponnusamy silvermann .
Kindly help!!