Complex Analysis/Complex Functions/Analytic Functions/Proof

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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.