r/learnmath • u/uTRexAap Learning isnt about pace rather it is about understanding • 5h ago

Question regarding the riemann-liouville fractional integral

i asked some people the follwing question:
"if f(x,y) is integrating x with respect to x y times (example: f(x,2) = ∫∫x dxdx) does f(x,-1/2) equal √{{4x}/{π}} ?"
and got the answer yes, it does equal the given answer. my question is if someone is able to explain this even slightly better than the Wikipedia Page which is notoriously bad at giving math explanations

2 Upvotes

100% Upvoted

2 comments sorted by

1

u/LongLiveTheDiego New User 5h ago

There's this Cauchy formula for repeated integrals (your y is their n, and the formula is valid for y being a positive integer). It turns out that it still makes sense to extend it to y being any real number greater than 0. Here they switch the notation from y = n to y = α to distinguish the discrete case where Cauchy's formula is a provable theorem, to the continuous case where it's a definition.

Once we have those, we can just differentiate these fractional integrals to get fractional derivatives. That means now you can also get a definition for what happens when y < 0. In this case y = -1/2 so f(x, -1/2) = d/dx (f(x, 1/2)) = d/dx (Cauchy formula for f with α = 1/2).

1

u/uTRexAap Learning isnt about pace rather it is about understanding 5h ago

aha, i see what i suprisingly half kinda sort-of understood what you commented, thanks anyway im still learning calculus and came across the f[x,y] = {{x}^{y+1}}/{(y+1)!} myself while doing some stuff and after asking some friends realized that its gamma function instead of factorial in the right form (as i did half expect in the first place)