What is Non-Integer Calculus and How Does it Work?

[meemoe_uk]

Seem to remember reading about RF experimenting with non-integer differentiation. I found it quite interesting to play with.
I started with 'half' differentiation.

e.g. f(x) = x^2 , D[1]f = 2x , D[2]f = 2
but what about non-integer diff?
e.g. D[0.5]f = px^1.5
what is p?

Clearly D[0.5]D[0.5]f = D[1]f and D[1](x^n) = nx^n-1
So D[0.5](x^n) = p(n)x^(n-0.5) and
D[0.5]( p(n)x^(n-0.5) ) = p(n)p(n-0.5)x^(n-1)
which is equal to D[1](x^n) = nx^(n-1)
so p(n)p(n-0.5) = n

From this I figure the key to finding p is solving the formula...
p(n)*p(n-0.5) = n .

It looks abit like the gamma function
G(n+1)/G(n) = n

and considering the solution to the gamma function, I decide that solving my non-integer diff formula is beyond me.
I've got a pretty good approximation though.

p ~= ((x+0.5)^0.5 + x^0.5)/2 for x>1

hmm. Is this 'genaralized' diff alreadly established math? Since calculus is so central to math,it seems to me that any generalization has broad applications.

 

[HallsofIvy]

Yes, it is established math. However, it is not done in that form because it can be shown that you can get the same results by using the standard calculus with different functions.

 

[M98Ranger]

Thank you for sharing your experience with non-integer calculus. It is indeed a fascinating topic that has been explored by mathematicians and scientists for many years. Non-integer calculus, also known as fractional calculus, deals with derivatives and integrals of non-integer orders, such as 0.5, 1.5, etc. This field has many applications in various fields, including physics, engineering, and economics.

Your approach to playing with half differentiation is interesting. It is true that the key to finding the value of p lies in solving the formula p(n)*p(n-0.5) = n. This formula is indeed similar to the gamma function, and the solution to it is known as the fractional gamma function. However, finding an exact solution to this formula is a challenging task and requires advanced mathematical techniques.

Many mathematicians have studied and developed generalized differentiation and integration techniques, including non-integer calculus. It has been established as a valid and useful branch of mathematics with broad applications. In fact, fractional calculus has been used to model various physical systems, such as viscoelastic materials, and to solve differential equations that cannot be solved using traditional calculus methods.

Thank you for bringing attention to this interesting topic. It is always exciting to see how mathematics can be extended and applied in new and innovative ways.