Cauchy-Riemann equations (complex coordinates)

Let f:G be a continuously differentiable function in the real sense, using 2 instead of , identifying f(z) with f(x,y) where z=x+iy and we also write z¯=x-iy (the complex conjugateMathworldPlanetmath). Then we have the following partial derivativesMathworldPlanetmath:

fz :=12(fx-ify),
fz¯ :=12(fx+ify).

Sometimes these are written as fz and fz¯ respectively.

The classical Cauchy-Riemann equationsMathworldPlanetmath are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to


This can be seen if we write f=u+iv for real valued u and v and then the differentialsMathworldPlanetmath become

fz =12(ux+vy)+i2(vx-uy),
fz¯ =12(ux-vy)+i2(vx+uy).

In several complex dimensions, for a function f:Gn which maps (z1,,zn)f(z1,,zn) where zj=xj+iyj we generalize simply by

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

Then the Cauchy-Riemann equations are given by

fz¯j=0  for all 1jn.

That is, f is holomorphic if and only if it satisfies the above equations.


  • 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Cauchy-Riemann equations (complex coordinates)
Canonical name CauchyRiemannEquationscomplexCoordinates
Date of creation 2013-03-22 14:24:28
Last modified on 2013-03-22 14:24:28
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 30E99
Related topic CauchyRiemannEquations
Related topic Holomorphic