Let Gn be a domain and let f:G be a C1 function (continuously differentiable) (z1,,zn)f(z1,,zn) where zj=xj+iyj. We can think of G as a subset of 2n. We therefore have the following partial derivativesMathworldPlanetmath for all 1jn,

fzj :=12(fxj-ifyj),
fz¯j :=12(fxj+ifyj).

Now let d be the standard exterior derivativeMathworldPlanetmath on 2n and the dxj and dyj the standard basis of cotangent vectors. Then if we define

dzj :=dxj+idyj,
dz¯j :=dxj-idyj,

then we can define two new operators acting on C1 functions on G giving 1-forms by

f :=j=1nfzjdzj,
¯f :=j=1nfz¯jdz¯j.

By direct calculation we immediately see that


Similarly we now define and ¯ on arbitrary differential form ω=α,βfα,βdzαdz¯β, where α and β range over all multi-indices with elements less then n, where if α=(α1,,αk) then dzα=dzα1dzαk, and fα,β is a C1, complex valued function on G.

ω :=α,βfα,βzjdzjdzαdz¯β,
¯ω :=α,βfα,βz¯jdz¯jdzαdz¯β.

Again a direct calculation shows that d=+¯.

The Cauchy-Riemann equationsMathworldPlanetmath are then given by


That is, f is holomorphic if and only if it satisfies the above equations. Note that this only applies to functions. If ¯ω=0 for a differential form, then the coefficients in the standard basis need not be holomorphic.


¯ and satisfy the following properties

  • ¯ and are linear,

  • ¯2=¯¯=0 and 2==0,

  • ¯-¯=0.

While ¯u=0 is our condition for u to be a holomorphic function it turns out that it is more important to solve the inhomogeneous ¯u=f equation, as that allows us to construct holomorphic objects from nonholomorphic ones.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title operator
Canonical name barpartialOperator
Date of creation 2013-03-22 15:10:39
Last modified on 2013-03-22 15:10:39
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 30E99
Classification msc 32A99
Synonym d bar operator
Synonym d-bar operator
Defines operator