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Explaining this notation may be as thankless a task as explaining a joke, but here goes.

It may help to think in terms of power series. Consider power series in two variables $x$ and $y$ with complex coefficients, instead of complex functions of real variables $x$ and $y$. Now note that such a thing can be rewritten as a power series in variables $z$ and $\bar z$ by the substitutions $x=\frac{1}{2}(z+\bar z)$ and $y=\frac{1}{2i}(z-\bar z)$. And you can convert back to the other form by the substitutions $z=x+iy$ and $\bar z=x-iy$. The holomorphic case is the case where the only terms in the $(z,\bar z)$ power series are the powers of $z$.

Whether you think in terms of power series or not, there is a fruitful fiction that complex functions of $x$ and $y$ can be thought of alternatively as functions of $z$ and $\bar z$, with the holomorphic ones being the functions of $z$ alone. This is consistent with the notation that you are asking about; see Qiaochu's comment to the question.

When I was a grad student the story went around that a fellow student attending the complex analysis course questioned precisely these signs, persistently suggesting that the professor had got them wrong. Finally the professor responded "Ah, I see! You and I must be thinking of different square roots of $-1$!"

(Edit: That's a joke.)

This same fellow student, when we were planning a skit for the annual math department picnic, objected to the title "Let Sleeping Dilogs" on the grounds that people who didn't know about dilogarithms wouldn't get the joke. He wanted to call it "Let Sleeping Dogs Lie" instead, which would have meant that there was no joke to get.

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Explaining this notation may be as thankless a task as explaining a joke, but here goes.

It may help to think in terms of power series. Consider power series in two variables $x$ and $y$ with complex coefficients, instead of complex functions of real variables $x$ and $y$. Now note that such a thing can be rewritten as a power series in variables $z$ and $\bar z$ by the substitutions $x=\frac{1}{2}(z+\bar z)$ and $y=\frac{1}{2i}(z-\bar z)$. And you can convert back to the other form by the substitutions $z=x+iy$ and $\bar z=x-iy$. The holomorphic case is the case where the only terms in the $(z,\bar z)$ power series are the powers of $z$.

Whether you think in terms of power series or not, there is a fruitful fiction that complex functions of $x$ and $y$ can be thought of alternatively as functions of $z$ and $\bar z$, with the holomorphic ones being the functions of $z$ alone. This is consistent with the notation that you are asking about; see Qiaochu's comment to the question.

When I was a grad student the story went around that a fellow student attending the complex analysis course questioned precisely these signs, persistently suggesting that the professor had got them wrong. Finally the professor responded "Ah, I see! You and I must be thinking of different square roots of $-1$!"

This same fellow student, when we were planning a skit for the annual math department picnic, objected to the title "Let Sleeping Dilogs" on the grounds that people who didn't know about dilogarithms wouldn't get the joke. He wanted to call it "Let Sleeping Dogs Lie" instead, which would have meant that there was no joke to get.