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1. Introduction
 
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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.


  Note: 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

(1.1)

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,

(1.2)

where

(1.3)

or equivalently,

(1.4)

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,

  1. 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).

  2. That the generalization of the factorial by the gamma function allows

    (1.5)

    which also is valid for non-integer values.

  3. The likeness of (1.2) to the binomial formula

    (1.6)

    which can be generalized to any complex number a by

    (1.7)

    which is convergent if

    (1.8)

There are some desirable properties that could be required to the fractional derivative,

  1. Existance and continuity for m times derivable functions, for any n which modulus is equal or less than m.

  2. 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.

  3. Iterating should not give problems,

    (1.9)

  4. Linearity,

    (1.10)

  5. Allowing Taylor's expansion in some other way.

  6. Its characteristic property should be preserved for the exponential function,

    (1.11)