关于“复数阶仿导数”和“复数阶微积分方程”的设想

The Ideas of "para-derivative of complex order"and "calculus equation of complex order

本文对于ｆ（ｘ）∈（Ｒ１），定义了它的＂复数阶仿导数ｆ（α）（ｘ）＂，并且对ｆ（ｘ）∈（Ｒ１）证明了ｆ（α）（ｘ）在全复平面上是复交量α的解析函数．我们发现当ａ＝－１时，ｆ＜１＞（ｘ）是ｆ（ｘ）的原函数，因此当Ｒｅα＜０时，我们又称ｆ（α）（ｘ）是ｆ（Ｘ）的复数阶积分．本文还对ｆ（ｘ）∈ｅ（Ｒ１）定义了它的＂复数阶广义仿导数ｆ（α）（ｘ）＂．文中还研究了方程．ｆ（α）（ｘ）－Ｇ（α）ｆ（ｘ）＝０（α∈Ｃ），并称之为＂复数阶微积分方程＂．得到了其解的表达式．

In this paper. firstly f(α) (x) (a∈C) is defined as f(x) ∈(R1) and called 'paraderivative of complex order'. Then f(α) (x) is proved to have analytic function for a ∈C when f(x) ∈D(R1). As f(α-1) (x) is found to have a primitive function of f<α> (x). so f(α) (x) is also called the complex order integral of f(x) if Reα < 0. After that, the meaning of f(α) (x) for f(x) ∈ D' (R1) is defined. Finally, a new kind of equation f(α) (x)-G(α)·f(x)=0 (named 'calculus equation of complex order') is considered and its solution is given in section 3.