In calculus we consider the idea where it is assumed that for certain such a limit is defined. An alternative notation is or . We may consider as an operator, let . Then . We can also say that .

As simple example, say that then we obtain that

(1.1)

therefore

(1.2)

Thus as is well-known. Now we may wonder, what is where is a positive integer and is any integer, it is clear that

(2)

So we may write

(3)

Now we use the definition of and we know that so we can extend the relation as

(4)

We now have a definition for the fractional operator , as acting on . Another example, consider then we obtain

(5.1)

Now we continue and we have

(5.2)

Therefore

(5.3)

as expected as . similar we find that

(6)

We now can consider something like and we find that

(7.1) so

(7.2) then

(7.3) therefore

(7.4)

Similar we have , for fractional calculus we have

(8.1)

(8.2)

(8.3)

(8.4) .

Now using the operators , we may also consider some operator like

(9) .

The is then fun part, as we have

(10.1) thus

(10.2)

Note the form – thus are eigenvectors and are eigenvalues.

Fractional calculus is an extension for normal calculus. The purpose of this post is simply to give some idea about fractional calculus. Have fun!