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Is it true that a function is analytic iff it satisfies the Cauchy-Riemann equations? I am reading Freitag's Complex Analysis and I am asked to show that ${\partial f\over \partial \bar{z}}=0$ iff $f$ is analytic. Is this because $f$ is analytic iff it satisfies the CR equations iff ${\partial f\over \partial \bar{z}}=0$? (It is obvious that $f$ satisfies CR $\implies {\partial f\over \partial \bar{z}}=0$ but what about the other relations? Are they true? I know that if a function is analytic, it must satisfy CR equations, but I don't know if the other direction is true or if ${\partial f\over \partial \bar{z}}=0$ necessarily mean that CR equations are satisfied.)

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2 Answers 2

To conclude from the Cauchy-Riemann equations or from $\frac{\partial f}{\partial \overline{z}}=0$ that $f$ is analytic, you probably want to assume at least that the partial derivatives of $f$ are continuous. It's possible to get along without that, but not at all easy: look up the Looman-Menchoff theorem.

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If $\frac{\partial f}{\partial \overline{z}}=0$, we can write $f=u(x,y)+iv(x,y)$ to get the following equations: $$ \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{1}{i}\frac{\partial}{\partial y}\right)f=0$$ and substituting our decomposition of $f$, $$ \frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x} +i\frac{\partial u}{\partial y}-\frac{\partial v}{\partial y}=0$$ From here, can you see that the Cauchy-Riemann equations must be satisfied?

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