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:

d²u/dx²=d/dx(dv/dy)=d/dy(dv/dx)=-d²u/dy²

Which proves that u is harmonic. Similar reasoning proves the same result for v, and thus f is harmonic.