This article completes the more than a half a century old problem of finding the equivalences between generalized Riemann derivatives. The real functions case is studied in a recent paper by the authors. The complex functions case developed here is more general and comes with numerous applications.

We say that a complex generalized Riemann derivative $\mathcal{A}$ implies another complex generalized Riemann derivative $\mathcal{B}$ if whenever a measurable complex function is $\mathcal{A}$-differentiable at z then it is $\mathcal{B}$-differentiable at z. We characterize all pairs $({\mathrm{\Delta }}_{\mathcal{A}},{\mathrm{\Delta }}_{\mathcal{B}})$ of complex generalized Riemann differences of any orders for which $\mathcal{A}$-differentiability implies $\mathcal{B}$-differentiability, and those for which $\mathcal{A}$-differentiability is equivalent to $\mathcal{B}$-differentiability. We show that all m points based generalized Riemann difference quotients of order n that Taylor approximate the ordinary nth derivative to highest rank form a projective variety of dimension $m-n$ for which an explicit parametrization is given.

One application provides an infinite number of equivalent ways to define analyticity. For example, a function f is analytic on a region Ω if and only if at each z in Ω, the limit$\underset{h\to 0}{\lim }\frac{f(z+h)+f(z+ih)-f(z-h)+f(z-ih)-2f(z)}{2h}$ exists and is a finite number. Four more applications relate the classification of complex generalized Riemann derivatives to analyticity and the Cauchy-Riemann equations, and to the theory of best approximations.

本文解决了一个半世纪以上的问题，即发现广义Riemann导数之间的等价关系。作者在最近的一篇论文中对实函数案例进行了研究。这里开发的复杂功能案例更为通用，并具有众多应用程序。

我们说一个复杂的广义黎曼导数 $\mathcal{一种}$ 意味着另一个复杂的广义黎曼导数 $\mathcal{乙}$ 如果每当有可测量的复杂函数 $\mathcal{一种}$-在z处可微，那么它是$\mathcal{乙}$-在z处可微分。我们表征所有对$（{\mathrm{\Delta }}_{\mathcal{一种}}，{\mathrm{\Delta }}_{\mathcal{乙}}）$ 任何阶的复杂广义黎曼差 $\mathcal{一种}$-可区分性意味着 $\mathcal{乙}$-可微性，以及那些 $\mathcal{一种}$-可区分性等于 $\mathcal{乙}$-可区分性。我们表明，所有米点数基于广义的顺序黎曼差商ň泰勒近似普通Ñ阶导数，以最高等级的形式投射各种尺寸的$米-ñ$ 为此，给出了明确的参数化。

一个应用程序提供了无数种等效的方法来定义分析性。例如，当且仅当在Ω中的每个z处有极限时，函数f才能在区域Ω上进行分析$\underset{H\to 0}{\mathrm{林}}\frac{F（ž+H）+F（ž+\mathrm{一世}H）-F（ž-H）+F（ž-\mathrm{一世}H）-\mathrm{2个}F（ž）}{\mathrm{2个}H}$存在并且是一个有限数。另外四个应用程序将复杂的广义Riemann导数的分类与解析度和Cauchy-Riemann方程以及最佳逼近理论相关。