Fractional Calculus is the branch of calculus that generalizes the derivative of a function to non-integer order, allowing calculations such as deriving a function to 1/2 order. Despite "generalized" would be a better option, the name "fractional" is used for denoting this kind of derivative.
These notes are not a summary of standard methods or an academic text, so some conventional developments are omitted while some other are introduced. The same happens with notation.
The derivative of a function f is defined as
Iterating this operation yields an expression for the n-st derivative of a function. As can be easily seen -and proved by induction- for any natural number n,
The case of n=0 can be included as well.
Such an expression could be valuable for instance in a simple program for plotting the n-st derivative of a function.
Viewing this expression one asks immediately if it can be generalized to any non-integer, real or complex number n. There are some reasons that can make us think so,
- The fact that for any natural number n the calculation of the n-st derivative is given by an explicit formula (1.2) or (1.4).
- That the generalization of the factorial by the gamma function allows
which also is valid for non-integer values.
- The likeness of (1.2) to the binomial formula
which can be generalized to any complex number a by
which is convergent if
There are some desirable properties that could be required to the fractional derivative,
- Existance and continuity for m times derivable functions, for any n which modulus is equal or less than m.
- For n=0 the result should be the function itself; for n>0 integer values it should be equal to the ordinary derivative and for n<0 integer values it should be equal to ordinary integration -regardless the integration constant.
- Iterating should not give problems,
- Allowing Taylor's expansion in some other way.
- Its characteristic property should be preserved for the exponential function,