Find all the analytic functions for which
$Im(z)=Re(f(z))$.
I know that a function is analytic in a domain if it is differentiable everywhere in that domain but I don't know where to go with the question.
I know that letting $z=x+iy$ then $Re(z)=x$ and $Im(z)=y$ but without a function to go on I don't know what $Re(f(z))$ is.
Maybe it helps to write $z = x+iy$ and $f(z) = u(x,y) + i v(x,y)$.
You want $Re(f(z)) = Im(z)$, so you must have $u(x,y) = y$.
Do you know the Cauchy-Riemann equations? This will allow you to determine $v$.