Thread: Complex Analysis Proof

Show that if a function f(z) = u(x,y) + iv(x,y) is entire, that the function conj(f(conj(z))) is entire. (Note that CR stands for Cauchy-Riemann)

Since f is entire, CR is satisfied:
so u_x = v_y and u_y = -v_x

this implies:
u_x = -(-v_y) and -u_y = -(-v_x)

this implies:
CR is satisfied for a function g(z) = u(x,-y) - iv(x,-y)

but g(z) = conj(f(conj(z)))

this implies:
CR is satisfied for conj(f(conj(z)))

Since f is entire, the partial derivatives are continuous

this implies:
conj(f(conj(z))) is entire.

That is correct.