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is the fractional derivative (if q > 0) or fractional integral (if q < 0). If q = 0, then the q-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several definitions of the differintegral.
The theory of fractional integration for periodic functions (therefore including the "boundary condition" of repeating after a period) is given by the Weyl integral. It is defined on Fourier series , and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the unit circle whose integrals evaluate to zero).
Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
Download as PDF; Printable version; From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Fractional calculus#Fractional integrals; Retrieved from "https
The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock–Kurzweil integrals. Specifically, if a continuous function F ( x ) admits a derivative f ( x ) at all but countably many points, then f ( x ) is Henstock–Kurzweil integrable and F ( b ) − F ( a ) is equal to the integral of f on [ a , b ] .
for < and >.. These are the fractional generalizations of the -fold left- and right-integrals of the form ()and for ,respectively. Even though the integral operators in question are close resemblance of the famous Erdélyi–Kober operator, it is not possible to obtain the Hadamard fractional integrals as a direct consequence of the Erdélyi–Kober operators.
If the differ integral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, information is lost, as with C in the first equation. However, in fractional calculus, given that the operator has been fractionalized and is thus continuous, an entire complementary function is needed.
The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:
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