相关论文: The Evaluation of a Quartic Integral via Schwinger…
In this paper we compute the 2-adic valuations of some polynomials associated with the definite integral $\int_{0}^{\infty} \frac{dx}{(x^4+2*a*x^2+1)^(m+1)}$
Closed form expressions are proposed for the Feynman integral $$ I_{D, m}(p,q) = \int\frac{d^my}{(2\pi)^m}\int\frac{d^Dx}{(2\pi)^D} \frac1{(x-p/2)^2+(y-q/2)^4} \frac1{(x+p/2)^2+(y+q/2)^4} $$ over $d=D+m$ dimensional space with…
Recently the Schur index of ${\cal N}=4$ SYM was evaluated in closed form to all orders including exponential corrections in the large $N$ expansion and for fixed finite $N$. This was achieved by identifying the matrix model which…
The method for analytic evaluation of four-particle integrals, proposed by Fromm and Hill, is generalized to include complex exponential parameters. An original procedure of numerical branch tracking for multiple valued functions is…
We use the method of brackets to evaluate quadratic and quartic type integrals. We recall the operational rules of the method and give examples to illustrate its working. The method is then used to evaluate the quadratic type integrals…
Analytic expressions for integrals which arise in a theory of atomic structure due to Schwinger and Englert are evaluated in terms of Bessel and Struve functions
We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.
A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.
The Moll-Arias de Reyna integral [1] $$\int_0^{\infty}\frac{dx}{(x^2+1)^{3/2}}\frac{1}{\sqrt{\varphi(x)+\sqrt{\varphi(x)}}}$$ $$\varphi(x)=1+\frac{4}{3}\left(\frac{x}{x^2+1}\right)^2$$ is generalised and several values are given.
We discuss a progress in calculations of Feynman integrals based on the Gegenbauer Polynomial Technique and the Differential Equation Method. We demonstrate the results for a class of two-point two-loop diagrams and the evaluation of most…
Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a…
We derive integral representations in terms of the Macdonald functions for the square modulus $s\mapsto | \Gamma ( a + i s ) |^2$ of the Gamma function and its Fourier transform when $a<0$ and $a\not= -1,-2,\ldots $, generalizing known…
Recasting the $N$-point one loop scalar integral from Feynman to Schwinger parameters gives an integrand with a Gaussian form. By application of a Fourier transform, it is easy to derive explicit expressions for the two, three and…
A proof is found for the elliptic integral evaluation of the Bessel moment $$M:=\int_0^\infty t I_0^2(t)K_0^2(t)K_0(2t) {\rm d}t ={1/12} {\bf K}(\sin(\pi/12)){\bf K}(\cos(\pi/12)) =\frac{\Gamma^6(\frac13)}{64\pi^22^{2/3}}$$ resulting from…
The integrals $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+1}\,e^{-\alpha r}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ and $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+2}\,e^{-\alpha r}\,j_\Llo(k_1r)\,…
One of the earliest examples of analytic representations for $\pi$ is given by an infinite product provided by Wallis in 1655. The modern literature often presents this evaluation based on the integral formula $$ \frac{2}{\pi} \int_0^\infty…
We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally…
This is a sequel to the paper [K. Fujii : SIGMA {\bf 7} (2011), 022, 12 pages]. In this paper we treat a non-Gaussian integral based on a quartic polynomial and make a mathematical experiment by use of MATHEMATICA whether the integral is…
We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The…
An analytical result is given for the exact evaluation of an integral which arises in the analysis of acoustic radiation from wave packet sources: $ I_{mn}(\beta,q) = \int_{-\infty}^{\infty} e^{-\beta^{2}x^{2}-i q x}x^{m+1/2}J_{n+1/2}(x)…