Content deleted Content added
→Real versus complex analytic functions: I had to explain this to a grad student in math the other day.Sigh. |
m →Real versus complex analytic functions: define ''harmonic'' |
||
Line 48:
In particular, note that for a complex coordinate ''z'', the complex function that gives the real part of ''z'', namely Re ''z'' = (''z'' + ''z''<sup>*</sup>)/2, is '''not''' analytic. This follows, as a complex analytic function cannot depend on the [[complex conjugate]]. This is true as well for the imaginary part Im ''z'' = -i (''z'' - ''z''<sup>*</sup>)/2. In particular, one does not get a real analytic function by taking the real part of a complex analytic function.
It also follows that complex analytic functions are [[Laplace's equation|harmonic]], whereas real analytic functions in general are not. That is, the [[Laplacian]], when applied to a complex analytic function, vanishes. This is most easily seen by writing the
:<math>\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} =
| |||