Suppose f(z)=u+iv is analytic. Under what circumstances will g(z)=u-iv be analytic?

Question

Hint: Consider the functions f(z)+g(z) and f(z)-g(z).

Answer 1

If you add the functions together, you get a purely real function. Suppose the function is analytic. Then we can differentiate from any direction, and in particular, from purely imaginary or purely real directions. The imaginary direction would give that, unless the derivative is zero, the function will have an imaginary derivative at every point. Similarly, from the real direction, the derivative must be purely real. Thus the derivative must be zero. It follows that $u$ must be constant. An analogous argument will show that $v$ must be constant, such that the only case is for $f(z)=c$ a constant such that $g(z)=\bar c$ is also a constant.