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The main idea is to express an integral involving an integer parameter (e.g. power) of a function, represented by I n, in terms of an integral that involves a lower value of the parameter (lower power) of that function, for example I n-1 or I n-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction ...
When the Abel–Plana formula is applied to the defining series of the polylogarithm, a Hermite-type integral representation results that is valid for all complex z and for all complex s: = + (, ) () + ( ) (+) / where Γ is the upper incomplete gamma-function.
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]
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
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 .
By means of integration by parts, a reduction formula can be obtained. Using the identity = , we have for all , = () () = . Integrating the second integral by parts, with:
Toyesh Prakash Sharma, Etisha Sharma, "Putting Forward Another Generalization Of The Class Of Exponential Integrals And Their Applications.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.2, pp.1-8, 2023.
In other words, it can solve these problems to a given accuracy in a number of operations that is proportional to the number of unknowns. Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid with a given grid point density .