Determine the complex function $f\left ( z \right )$

Question

The details provided are that the function is analytic and that its real part along the line $y=c x$ is constant. What conclusions can I draw from here? I think that this imposes $\frac{\partial u}{\partial x}=0 $ and $\frac{\partial u}{\partial y}=0 $ along the line, however I am unsure how to take it from here.

Answer 1

Consider the function $g(z) := \mathrm i (f((1+\mathrm ic)z)-f(0))$. Clearly $g$ is analytic iff $f$ is analytic. Moreover, $(1+\mathrm ic)z$ has the property that $y=cx$ iff $z\in\mathbb R$. Finally, for real $z$, clearly $g(z)$ has a zero imaginary part, that is, it is real.

So $g(x)$ can be any analytic function that maps real values to real values.

Answer 2

Look at $f(z)=z^2=x^2-y^2+2ixy$. We have $u(x,y)=x^2-y^2$ and $v(x,y)=2xy$. Take $c=1$. Along the line $y=x$, we have $u=0$, but the partial derivatives of $u$ are not $0$ on the line.