Cauchy–Riemann equations
Complex differential forms

As often formulated, the d-bar operator ∂ ¯ {\displaystyle {\bar {\partial }}} annihilates holomorphic functions. This generalizes most directly the formulation ∂ f ∂ z ¯ = 0, {\displaystyle {\partial f \over \partial {\bar {z}}}=0,} where ∂ f ∂ z ¯ = 1 2 (∂ f ∂ x + i ∂ f ∂ y). {\displaystyle {\partial f \over \partial {\bar {z}}}={1 \over 2}\left({\partial f \over \partial x}+i{\partial f \over \partial y}\right).}
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