Let's take some function

and some fractional differentiation method

such that

Now create the function:

Integrate by parts, and for

we get the spectacular identity that:

What is going on here?

(08/22/2013, 05:54 PM)JmsNxn Wrote: [ -> ]

Integrate by parts, and for we get the spectacular identity that:

I do not even need to use integrate by parts to see a problem.

Its funny you say

because its more like an equality when we differentiate a given amount of times with respect to t.

You see : s is considered a constant with respect to t since s is not a function OF t NOR f.

There is big difference between a function , an operator , a variable and a constant.

ALthough that may sound belittling or trivial , your example shows this is an important concept !!

If you consider

as a function F(s,f) then it is no surprise that taking the derivative with respect to f leaves s unchanged.

By the chain rule you then get the " wrong " / " correct "

if you take the derivative

times.

This is similar to

.

Hence by the very definition of the gamma function you also get

here which you already showed yourself with the - overkill - method integrate by parts.

This might not answer all your questions yet but I assume it helps.

It not completely formal either sorry.

It might affect your other posts about integral representations for fractional calculus , tetration and continuum sum.

Im still optimistic though and hope I did not discourage you to much.

regards

tommy1729

I just read that over today and it makes a lot more sense a second time through.

I've been finding a lot of interesting paradoxes with fractional calculus and it must be my lack of rigor. This one and the function which if converges is its own derivative:

i.e:

OH! That makes a lot of sense. That's very interesting.