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Published January 8, 2026 | Version v1
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Method for the Exact Analytical Representation and Evaluation of Special Functions Defined by Non-Elementary Integrals via Cauchy's Mean Value Theorem

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Description

This paper introduces a novel computational and analytical framework for representing non-elementary integrals through the application of Cauchy's Mean Value Theorem. By establishing a functional relationship between a target integral

f(x)f of x
𝑓(π‘₯)

and a selected auxiliary elementary function

g(x)g of x
𝑔(π‘₯)

, we demonstrate that the value of the integral

f(b)f of b
𝑓(𝑏)

can be exactly determined at a specific mean point

c∈(0,b)c is an element of open paren 0 comma b close paren
𝑐∈(0,𝑏)

. This point

cc
𝑐

is derived via the Lagrangian geometric projection of the chord. A comprehensive catalog of auxiliary function pairs for fundamental special functions, including Integral Sine, Error Function, and Fresnel Integrals, is provided, utilizing the

A(t)cap A open paren t close paren
𝐴(𝑑)

 integrand representation.

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Dates

Copyrighted
2026-01-08