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 :
⁂
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.
⁂
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
