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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).
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.
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.
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 ...
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:
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
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.