Complex Analysis/Complex Functions/Analytic Functions/Proof
Namely if the function is analytic its real and imaginary parts must have the partial derivative of all orders the function is analytic it must satifiy the Cauchy Riemann equation.
A holomorphic function is harmonic, provided it is of class C2
Let the function f = u+iv be holomorphic and of class C2.
By the Cauchy-Riemann equations, we have:
Which proves that u is harmonic. Similar reasoning proves the same result for v, and thus f is harmonic.