## Cauchy-Riemann

Friday 28 October 2005 at 2:49 pm | In Articles | 1 Comment

The Cauchy-Riemann equations are one of the first results one comes across in Complex Analysis. A poster on S.O.S. Mathematics Cyberboard has pointed that that proofs like that at Cauchy-Riemann equations tend to take it for granted that if is analytic then the partial derivatives of and exist. Thus the proof at Cauchy-Riemann equations says

and then deduces that

Looking at various textbooks this omission seems to the norm. Even Ahlfors Complex Analysis says: We remark that the existence of the … partial derivatives … is implied by the existence of

One excellent book A First Course in Complex Functions by G.J.O. Jameson does give a proper proof of this result. It defines differentiability for (where is a subset of ) at a point in the interior of if there exists real numbers such that, given , there exists such that, for all real with ,

Putting shows that ; similarly

If then, given , there exists such that for all real with

and taking real parts

from which it follows that and exist. Taking imaginary parts gives the other 2 partial derivatives.

## 1 Comment »

1. I think that both the poster at SOS and you missed the point here. If exists than it follows easily that the partials exist by the fact that if one restricts oneselves to real h, than the existence of implies that the real part of the righthand side has as limit the real part of (and thus the partial of u wrt x exists). This follows through the inequality . Similarly for the imaginary part.

Comment by Anton R. Schep — Saturday 12 November 2005 2:22 am #