柯西黎曼条件为什么这么神奇? 第1页

Well, real-valued analytic functions are just as rigid as their complex-valued counterparts. The true question is why complex smooth (or complex differentiable) functions are automatically complex analytic, whilst real smooth (or real differentiable) functions need not be real analytic.

As Qiaochu says, one answer is elliptic regularity: complex differentiable functions obey a non-trivial equation (the Cauchy-Riemann equation) which implies a integral representation (the Cauchy integral formula) which then implies analyticity (Taylor expansion of the Cauchy kernel); the ellipticity of the Cauchy-Riemann equation is what gives the analyticity of its fundamental solution, the Cauchy kernel. Real differentiable functions obey no such equation.

Another approach is via Cauchy's theorem. In both the real and complex setting, differentiability implies that the integral over a closed (or more precisely, exact) contour is zero. But in the real case this conclusion has trivial content because all closed contours are degenerate in one (topological) dimension. In the complex case we have non-trivial closed contours, and this makes all the difference.

EDIT: Actually, the above two answers are basically equivalent; the latter is basically the integral form of the former (Morera's theorem). Also, to be truly nitpicky, "differentiable" should be "continuously differentiable" in the above discussion.

其问题是

Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)
Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?

跟题主的问题有些不同，但可以一看.

引自mathoverflow，请读者务必点进去看……因为回答者是User Terry Tao

说一下自己的看法。

首先，一个函数满足一个微分方程这件事情其实挺不平凡的，以一维为例，随便写一个二阶的微分方程，比如说

，解出来一定是光滑的。所以个人感觉，一个微分方程蕴含很多信息这件事情其实挺常见的对吧。

然后，关于elliptic differential operator的regularity，说实话这件事情其实挺复杂的，可以从Sobolev空间的先验估计去看，这个需要一些细致的计算太不直观了，或者从pseudo differential operator的parametrix来看， 这个看法稍微直观一点，但是关于PDO是什么这件事情还是需要一些预备知识的，比如Fourier analysis，还有看parametrix也可以从functional calculus来看，还有PDO的formal development，当然还有最关键的，Sobolev space的embedding theorem……总之……一言难尽……

但是，说回来holomorphic这件事情，个人感觉还是从Cauchy integral来看更好，因为CR方程不仅仅说的是光滑，还有解析，光滑和解析差的太多太多了，仅仅光滑是不会导出全纯函数的那么多好性质的。而解析的话归根结底是因为CR方程的integral kernel是解析的，大概就这样。