Let f be a function such that: $$f(x+iy) = u(x) + iv(y)$$ (that is, such that u does not depend on y and v does not depend on x). Prove that if f is analytic in C then $f(z) = az + b$ for some a, b elements of C.
As f is analytic it satisfies the Cauchy Riemann equations.
$u_x = v_y$, $u_y = -v_x$
By definition $f'(z) = u_x + iv_x$
but $v_x = 0$ so
$f'(z) = u_x$
Now $u_x$ is just some element of C. Call it a.
$f'(z) = a$
Integrate both sides with respect to z and we get
f(z) = az + b
Does this look right? Particularly the part where I let $u_x = a$?