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p_g=partial diff

_p_g=conjugate pde

p_g_x=partial diff with respect to x

Suppose that f : D → C . Let g(z) : D → C be defined by

g(z) = f(_z). Calculate_p_g and p_g where _p_g=(p_g_x+i(p_g_y))/2, p_g=(p_g_x+i(p_g_y))/2;

Conclude that f is holomorphic on D if

and only if p_g = 0 on D.

I've calculated the pde and observed that I get different signs that supposed to for a regular exercise.Also there is a property that states that f is holomorphic if _p_g=0.....given the nature of the function g(z) by intuition am assuming that in this case is has to be the other way around.

Any help will be great Thank you.