## What are the application of the integral and differential in times from integer to rational ,real and complex [closed]

For instance,if $\int f=\frac{1}{a}e^{ax}$ is regard as integral in one time,we use notation $T(1)\int f=\frac{1}{a}e^{ax}$,we may extend it to fractional ,real,or complex time in such a way:$$T(\alpha)\int f=\frac{1}{a^{\alpha}}e^{ax},\alpha \in N,Z,Q,R,or C$$.

We even may extend $\alpha$ to a more general field than C.

First question :with what class of function f can the intergral be extended in such a way? every integralable one?

Second question:what is the application of such extension in math?Anyone gives any example?

Third question:can we explain such extension or operation by geometry as usual integral?How should we explain such extension or operation by geometry?It can not intepreted in geometry?

Fourth question:what are the more general extension of such operation than C?

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 I'm afraid I cannot understand what you mean by extending the function $f$. Your Second and third questions are also far too unfocused. Please try to reformulate your question in a more mathematically precise way. – Yemon Choi Oct 30 2011 at 23:16 The Fractional Calculus (en.wikipedia.org/wiki/Fractional_calculus) may be what's behind this... but you should turn this question into once that is more clearlyu understandable. – Mariano Suárez-Alvarez Oct 30 2011 at 23:36 @Suarez-Alvarez,it can be extended or generalized to real and complex,so I think there are difference between fractional calculus and the generalization.Having read the link you give me ,I think so.Anyway,thank you for your useful comment.And I ask if it can be extended to more general field than C – XL Oct 30 2011 at 23:44