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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.
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 k {\displaystyle k} previous terms of the sequence appear in the equation, for a parameter k {\displaystyle k} that is independent of n {\displaystyle n} ; this number k ...
The integral here is a complex contour integral which is path-independent because is holomorphic on the whole complex plane . In many applications, the function argument is a real number, in which case the function value is also real.
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.
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.
Integral equations are important in many applications. Problems in which integral equations are encountered include radiative transfer, and the oscillation of a string, membrane, or axle. Oscillation problems may also be solved as differential equations. Actuarial science (ruin theory [8]) Computational electromagnetics
The sequence () is decreasing and has positive terms. In fact, for all : >, because it is an integral of a non-negative continuous function which is not identically zero; + = + = () () >, again because the last integral is of a non-negative continuous function.
The same formula is then used to compare the four piece and the two piece estimate, and likewise for the higher estimates; For the second iteration the values of the first iteration are used in the formula 16 × (more accurate) − (less accurate) / 15