Saturday, September 14, 2024

HP Prime CAS: Riemann-Louiville Integral vs Taking the Indefinite Integral Twice

HP Prime CAS: Riemann-Louiville Integral vs Taking the Indefinite Integral Twice



Introduction


The Riemann-Louiville Integral takes the integral of the function f(x) of any positive order v. The integral is defined as:


c_D_x^(-v) = 1 / Γ(v) * ∫( (x – t) * f(t) dt, t = c, t = x)


where:

c = a real constant, which can be zero

f(x) = function of x

t = dummy variable of integration

‘v = order where v >0


If v=1, this is the regular integral. However, the value of v can be a positive non-integer. If v=2, then the Riemann-Louiville integral is a result if you integrate the function twice.


This is one of the formulas on determining indefinite integrals of various orders.



HP Prime CAS Function: dblint


Double Integral of f(x) which takes the integral of f(x) twice. The variable x is used in the function.


dblint(f):= ∫∫ f dx dx


HP Prime CAS Function: rlint


The Riemann-Liouville Integral of f(x). The input has the variable x as the independent variable. The result of the function returns t as the independent variable.


rlint(f,c,v):=(∫(t–x)^(v–1)*f,x,c,t)) / Gamma(v)


Note that the variables x and t are switched to allow the input to be a function of x.


Notes


This was programmed on the CAS page in the format:


func(var) := function


I was not able to use the Program Editor mode at time of programming (July 30, 2024). (Beta Firmware 15048)


Examples



Double Integral: dblint

RLI, v = 2: rlint with c = 0

f(x) = x^m, m>0

x^(m+2) / (m^2 + 3*m + 2)

t^(m+2) / (t^2 + 3*t + 2)

f(x) = a*x + b

(a*x^3 + 3*b*x^2) / 6

(a*t^3 + 3*b*t^2) / 6

f(x) = e^x

e^x

-t + e^t – 1

f(x) = cos x

-cos x

-cos t + 1


Taking the derivative twice will return us back to the original function. Note that the indefinite integral function assumes that the added constant is zero ( ∫ f(x) dx = F(x) + C ).



Source


Kimeu, Joseph M., "Fractional Calculus: Definitions and Applications" (2009).Masters Theses & Specialist Projects. Paper 115. http://digitalcommons.wku.edu/theses/115



Eddie


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