Generalizations of Calculus I

This is the beginning of a series of posts on this subject. I'll stick to the basic stuff and go from here, a primer if you will.

Calculus is a wonderful topic and I'd like to assume you know enough about it. feel free to email me or comment if you think i said anything wrong or if you don't understand anything.

Everybody knows (everybody who matters) about the operation of differentiation. lets call it D

D^1(f(x))= d/dx(f(x))
and successive iterations of these function are taken we obtain a function of two variables that is the nth derivative of a function 'f' at the value 'x'.

The first derivative of a well-behaved-single-real-variable function can be understood as the slope of the line formed by the ordered pair (x, f(x)), or f'(x). the second derivative is a measure of the concavity of said function. each succesive derivative represents the rate of change of the rate of change of the rate of change of the rate of change.....n of a function x with respect to x.

n does not have to be a positive number. it could be a negative one. the operator can simultaneously include its inverse operator and antiderivatives can just be derivatives of negative order. so the operator D^[Z](f(x)) can represent the fundamental operations of calculus: integration and differentiation as a single process and the successive repetitions of them in the same way.

There is however one problem.

Derivatives lose information and integrals require information.

a function f(x) := g(x) + c where c is a constant independent of x will have derivative f'(x)= g'(x) and extra information will be required to reverse the process to the initial state. more so if this process is repeted several times. an Nth order integral will be indeterminate to (N-1) polynomial order unless this information is provided. this problem is hurtful to the generalization of the differintegral.

but for now I'll give you one of these generalizations
this is Cauchy's formula for repeated integration

this is the Riemann-Liouville differintegral


which generalizes derivatives. please note the difference in variables .
(will fix this later)

Think about that