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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.
In mathematics, the Weyl integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions f on the unit circle having integral 0 and a Fourier series. In other words there is a Fourier series for f of the form
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).
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.
In mathematics, an Erdélyi–Kober operator is a fractional integration operation introduced by Arthur Erdélyi and Hermann Kober . The Erdélyi–Kober fractional integral is given by x − ν − α + 1 Γ ( α ) ∫ 0 x ( t − x ) α − 1 t − α − ν f ( t ) d t {\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0 ...
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:
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field.
A fractional-order integrator or just simply fractional integrator is an integrator device that calculates the fractional-order integral or derivative (usually called a differintegral) of an input. Differentiation or integration is a real or complex parameter.
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