Show that a harmonic function has partial derivatives of all orders.

Question

I have a corollary in my textbook that says "If f(z) is analytic on a domain D, then f(z) is infinitely differentiable, and the successive complex derivatives f'(z), f''(z)...., are all analytic on D. I know I can use this corollary if I can show that u(z) has a harmonic conjugate, but I'm not sure how to go about doing it.

Answer 1

The derivative of a holomorphic function $f(x+iy) = u(x,y) + i v(x,y)$ is given by the partial derivatives of its real and imaginary parts: $f'(x+iy) = u_x(x,y) + i v_x(x,y)$. Now you can use the Cauchy-Riemann equations to get what you are after.

Answer 2

any harmonic function is the real part of an analytic function , so we simply can find v(x,y) what is conjuctive in d , to u(x,y) since v(x,y) is harmonic conjugate to u(x,y) , f(z)= u+ iv is analytic

hence , every analytic function it's derivates of all orders are analytic.