Calculation of integrals in MathPartner

We present the possibilities provided by the MathPartner service of calculating definite and indefinite integrals. MathPartner contains software implementation of the Risch algorithm and provides users with the ability to compute antiderivatives for elementary functions. Certain integrals, including improper integrals, can be calculated using numerical algorithms. In this case, every user has the ability to indicate the required accuracy with which he needs to know the numerical value of the integral. We highlight special functions allowing us to calculate complete elliptic integrals. These include functions for calculating the arithmetic-geometric mean and the geometric-harmonic mean, which allow us to calculate the complete elliptic integrals of the first kind. The set also includes the modified arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. The Lagutinski algorithm is of particular interest. For given differentiation in the field of bivariate rational functions, one can decide whether there exists a rational integral. The algorithm is based on calculating the Lagutinski determinant. This year we are celebrating 150th anniversary of Mikhail Lagutinski.

Simple Closed Analytic Formulas to approximate the Legendre Complete Elliptic Integrals K(k) and E(k), by a Fast Converging Recurrent-Iterative Procedure

wo sets of closed analytic functions are proposed for the approximate calculus of the complete elliptic integrals K(k) and E(k) in the normal form due to Legendre, their expressions having a remarkable simplicity and accuracy. The special usefulness of the newly proposed formulas consists in they allow performing the analytic study of variation of the functions in which they appear, using derivatives (they being expressed in terms of elementary functions only, without any special function; this would mean replacing one difficulty by another of the same kind). Comparative tables of so found approximate values with the exact ones, reproduced from special functions tables, are given (vs. the elliptic integrals’ modulus k). Both sets of formulas are given neither by spline nor by regression functions. The new functions and their derivatives coincide with the exact ones at the left domain’s end only. As for their simplicity, the formulas in k / k' do not need mathematical tables (are purely algebraic). As for accuracy, the 2nd set, more intricate, gives more accurate values and extends itself more closely to the right domain’s end. An original fast converging recurrent-iterative scheme to get sets of formulas with the desired accuracy is given in appendix.

Fishnet four-point integrals: integrable representations and thermodynamic limits

We consider four-point integrals arising in the planar limit of the conformal “fishnet” theory in four dimensions. They define a two-parameter family of higher-loop Feynman integrals, which extend the series of ladder integrals and were argued, based on integrability and analyticity, to admit matrix-model-like integral and determinantal representations. In this paper, we prove the equivalence of all these representations using exact summation and integration techniques. We then analyze the large-order behaviour, corresponding to the thermodynamic limit of a large fishnet graph. The saddle-point equations are found to match known two-cut singular equations arising in matrix models, enabling us to obtain a concise parametric expression for the free-energy density in terms of complete elliptic integrals. Interestingly, the latter depends non-trivially on the fishnet aspect ratio and differs from a scaling formula due to Zamolodchikov for large periodic fishnets, suggesting a strong sensitivity to the boundary conditions. We also find an intriguing connection between the saddle-point equation and the equation describing the Frolov-Tseytlin spinning string in AdS3 × S1, in a generalized scaling combining the thermodynamic and short-distance limits.

Closed Analytic Formulas for the Approximation of the Legendre Complete Elliptic Integrals of the First and Second Kinds

The author proposes two sets of closedanalytic functions for the approximate calculus of thecomplete elliptic integrals of the first and secondkinds in the normal form due to Legendre, therespective expressions having a remarkablesimplicity and accuracy. The special usefulness of theproposed formulas consists in that they allowperforming the analytic study of variation of thefunctions in which they appear, by using thederivatives. Comparative tables including theapproximate values obtained by applying the two setsof formulas and the exact values, reproduced fromspecial functions tables are given (all versus therespective elliptic integrals modulus, k = sin ). It is tobe noticed that both sets of approximate formulas aregiven neither by spline nor by regression functions,but by asymptotic expansions, the identity with theexact functions being accomplished for the left end k= 0 ( = 0) of the domain. As one can see, the secondset of functions, although something more intricate,gives more accurate values than the first one andextends itself more closely to the right end k = 1 ( =90) of the domain. For reasons of accuracy, it isrecommended to use the first set until  = 70.5 only,and if it is necessary a better accuracy or a greaterupper limit of the validity domain, to use the secondset, but on no account beyond  = 88.2.

Master integrals for bipartite cuts of three-loop photon self energy

We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

Some Padé approximations and inequalities for the complete elliptic integrals of the first kind

In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind $K(x)$ K ( x ) , and motivated by these approximations we also present the following double inequality: $$ \frac{1-x^{2}}{1-x^{2}+\frac{x^{4}}{62}}< \frac{2 e^{\frac{2}{\pi }K(x)-1}}{ (1+\frac{1}{\sqrt{1-x^{2}}} )}< \frac{1-\frac{96}{100}x^{2}}{1-\frac{96}{100}x^{2}+\frac{x^{4}}{64}},\quad x\in ( 0,1 ). $$ 1 − x 2 1 − x 2 + x 4 62 < 2 e 2 π K ( x ) − 1 ( 1 + 1 1 − x 2 ) < 1 − 96 100 x 2 1 − 96 100 x 2 + x 4 64 , x ∈ ( 0 , 1 ) . Our results have superiority over some new recent results.