Gold trader. The Relationship Between Even Powers of the Gaussian Integral, the Gamma Function, and Fibonacci Levels

© Vitaly Fartushnov, 2025

ISBN 978-5-0067-7881-8

Создано в интеллектуальной издательской системе Ridero

Even powers of the Gauss integral connection with the gamma function

The relationship between even powers of the Gauss integral and the gamma function is expressed as follows:

Let us consider an integral of the form

where is an even degree.

This integral is expressed in terms of the gamma function as follows:

or, using the properties of the gamma function:

Replacing the variable $ t = x^2 $ leads to the gamma function:

Thus,

– Even moments of the Gaussian integral are expressed through the gamma function of a half-integer argument.

– For the general case: where for standard Gauss $ m=2n,,a=1,,b=2 $ [1] [2].

This shows the direct connection between even powers of the Gauss integral and the values of the gamma function, and explains why the values for half-integer arguments are expressed in terms of $ \sqrt {\pi} $ [1] [2].

– For example, for $ n=0 $ (that is, the standard Gauss integral):

– For $ n=1 $:

That is, the values of even Gaussian moment integrals are directly expressed through the gamma function for half-integer arguments [1] [2].

Examples of use :

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List the literature on this topic

The main sources on the topic of Gauss integrals with even powers and related topics of quadratures, calculations and applications are:

– Fichtenholz G.M. «Course of Differential and Integral Calculus», v.2. Classic textbook, examines in detail Gaussian integrals, their properties and calculations ^>[3].

– Kraynov V.A. «Numerical Methods for Calculating a Definite Integral». The manual examines methods for exact and approximate calculation of integrals, including Gaussian integrals, discusses accuracy for even-degree polynomials, and provides formulas ^>[4].

– Fedotov AA «Numerical methods of integration». Modern manual with practical examples of numerical calculation of Gaussian integrals and their applications ^>[5].

– Pastukhov D.F. «Gauss quadratures. Integral equations». Methodical instructions for the construction and application of Gauss quadrature formulas for calculating integrals of polynomials even of high degree ^>[6].

– Dobrynina S.V. «Quadrature and cubature formulas». The manual contains practical tasks and analysis of the rules of approximate integration, including Gauss’s formulas for integrals with weights and polynomials ^>[7].

– Arushanyan I.O. «Algorithms for approximate calculation of integrals». Methods for calculating integrals with various degrees of accuracy are discussed, including the use of even powers in Gaussian integrals ^>[8].

These books and manuals are suitable for an in-depth study of the calculation and theory of Gaussian integrals, their properties and applications.

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Theoretical foundations of quadrature formulas

Quadrature formulas are a classic mathematical tool for the approximate calculation of definite integrals over function values at a finite number of points. Their main purpose is to replace the calculation of the integral with a sum, where the coefficients and choice points (nodes) are selected to achieve maximum accuracy on a certain class of functions [9] [10] [11].

General form of the quadrature formula

Any quadrature formula usually has the following form:

Where:

– — formula nodes (points at which the function is calculated),

– — weights (coefficients with which the function value at the nodes is taken).

Theoretical Provisions

– The accuracy of a quadrature formula is determined by the class of functions for which the formula gives the exact value of the integral. Usually these are polynomials of degree no higher than a certain value.

– The error (remainder term) of the quadrature formula shows how accurately the integral is approximated [10] [12].

– Selection of nodes and weights: For example, in the Newton—Cotes formulas, the nodes are equidistant, while in the Gauss formulas, they are the roots of orthogonal polynomials (for example, Legendre polynomials for the classical integral on a segment), which allows achieving the highest algebraic degree of accuracy – for nodes ^>[11].

– Historical background: methods of approximate integration arose in the times of Newton and Leibniz. The construction of classical formulas (rectangles, trapezoids, Simpson) is based on replacing the function inside the segment with some simple approximation (polynomial) and integrating this approximation ^>[10].

Types of quadrature formulas

– Interpolation quadrature formulas: are based on the interpolation of a function (usually a polynomial one) ^>[11].

– Compound (interval partitioning): a long interval is divided into parts, and a simple formula (such as the trapezoid or Simpson compound formula) is applied to each part [12] [11].

– Weighted quadrature formulas are used when the integrand contains singularities (e.g., singularities or rapid growth) [10] [13].

Error and degree of accuracy

– The degree of accuracy is the highest degree of a polynomial for which the formula gives an exact integral.

– The error is estimated through the upper bound of the high-order derivatives of the integrand (usually the Taylor residue formula) [11] [10] [9].