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In integral calculus, integration by reduction formulae is a method relying on recurrence relations. It is used when an expression containing an integer parameter , usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree , can't be integrated directly.
By means of integration by parts, a reduction formula can be obtained. ... This is a recurrence relation giving in terms of . This, together ...
The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antiderivatives of a function into a single integral (cf. Cauchy's formula). For non-integer n it yields the definition of fractional integrals and (with n < 0) fractional derivatives .
An integral recurrence that is deduced and demonstrated in [6] ... generating function by using Cauchy's integral formula to write the Hermite polynomials as = ...
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
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.
The integral is Euler's harmonic number, so the previous formula may also be written (+) = +. A consequence is the following generalization of the recurrence relation ...
These numbers are named after James Gregory who introduced them in 1670 in the numerical integration context. They were subsequently rediscovered by many mathematicians and often appear in works of modern authors, who do not always recognize them. [1] [5] [14] [15] [16] [17]