Study Notes For Fractional Calculus II
Differintegrals for Functions
Proof of Differintegral Power Rule
Let
Then by standard calculus, we can have the following relationships,
By extending the order of the integral to a real number, we can have the following:
We can see that the derivative formula we derived here using integration formula matches the fractional derivative formula presented at the very begining of {% post_link Study-Notes-For-Fractional-Calculus-I ‘Note I’ %}. Now we want to give a formal proof that the intuitive “fractional derivative power rule” we get in the previous note is actually true by using R-L Integral.
Want to show:
Proof:
Therefore,
At this step, we can see that the fractional power rule we derived from our intuitive guess do matches with our theory.
Differintegrals for Exponential Functions
Differintegrals for Sine and Cosine
And intuitive way for deriving differintegral for sine function can be taking the taylor expansion of the sine function and apply differintegral power rule to each term in the formula. As our differintegral power rule has base point 0, the formula for sine can be derived as following:
Do a few examples using the formula derived above:
Ex1:
Ex2:
Ex3:
We can see that deriving formula for sine function using taylor exapansion will make the result function no longer periodic, which is an undesired behavior. Therefore, we want to derive the formula in another way.
In standard calculus, we have the following relationships:
We now want to show if the formula we derived from intuition is actually the case. To do that, we need to findout a proper value for the base point
Short Summary
For now, we have derived differintegrals for the following functions:
For the fractional power rule, we do not have the same good property, having
Stickers by the end of this note
- Using numerical trick for showing the validity of the differintegral formula for trignometry is evil, need to derive concrete a more concrete proof.
- Differintegrals over sine and cosine are basically shifting the phase of the function, this seems to be a really good property and we might be able to use this property to quickly derive the differintegral of any function using Fourier Transform.
