Cauchy-Riemann equations (complex coordinates)
Let f:G⊂ℂ→ℂ be a continuously differentiable function in the real sense, using ℝ2 instead of
ℂ, identifying f(z) with f(x,y) where z=x+iy and we also write ˉz=x-iy (the complex conjugate). Then we have the following partial derivatives
:
∂f∂z | := | ||
Sometimes these are written as and respectively.
The classical Cauchy-Riemann equations are equivalent
to
This can be seen if we write for real valued and and
then the differentials become
In several complex dimensions, for a function which maps where we generalize simply by
Then the Cauchy-Riemann equations are given by
That is, is holomorphic if and only if it satisfies the above equations.
References
- 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 |