Photon boomerang in a nearly extreme Kerr metric

The Kerr rotating black hole metric has unstable photon orbits that orbit around the hole at fixed values of the Boyer-Lindquist coordinate r that depend on the axial angular momentum of the orbit, as well as on the parameters of the hole. For zero orbital axial angular momentum, these orbits cross the rotational axes at a fixed value of r that depends on the mass M and angular momentum J of the black hole. Nonzero angular momentum of the hole causes the photon orbit to rotate so that its direction when crossing the north polar axis changes from one crossing to the next by an angle I shall call ∆φ, which depends on the black hole dimensionless rotation parameter a/M = cJ/(GM2) by an equation involving a complete elliptic integral of the first kind. When the black hole has a/M ≈ 0.994 341 179 923 26, which is nearly maximally rotating, a photon sent out in a constant-r direction from the north polar axis at r ≈ 2.423 776 210 035 73 GM/c2returns to the north polar axis in precisely the opposite direction (in a frame nonrotating with respect to the distant stars), a photon boomerang.

Total cross sections of eγ → $$ eX\overline{X} $$ processes with X = μ, γ, e via multiloop methods

Using modern multiloop calculation methods, we derive the analytical expressions for the total cross sections of the processes e−γ →$$ {e}^{-}X\overline{X} $$ e − X X ¯ with X = μ, γ or e at arbitrary energies. For the first two processes our results are expressed via classical polylogarithms. The cross section of e−γ → e−e−e+ is represented as a one-fold integral of complete elliptic integral K and logarithms. Using our results, we calculate the threshold and high-energy asymptotics and compare them with available results.

Concise high precision approximation for the complete elliptic integral of the first kind

<abstract><p>In this paper, we obtain a concise high-precision approximation for $ \mathcal{K}(r) $:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \frac{2}{\pi }\mathcal{K}(r){\rm{ }}&gt;{\rm{ }}\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>which holds for all $ r\in (0, 1) $, where $ \mathcal{K}(r) $ is complete elliptic integral of the first kind and $ r^{\prime } = \sqrt{1-r^{2}} $.</p></abstract>

Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants representable as contiguous very well-poised generalized hypergeometric series and as even moments of the square of the complete elliptic integral of the first kind. Neither constants are currently available in closed form. New symmetries are found in the critical values of the L-series of two underlying cusp forms, providing a sense in which one of the constants has a formal counterpart. The significant roles this constant and its counterpart play in multidisciplinary contexts is described. The results unblock the problem of representing them in terms of lower-order generalized hypergeometric series, offering progress towards identifying their closed forms. The same approach facilitates a canonical characterization of the hypergeometry of the parbelos, adding to the characterizations outlined by Campbell, D'Aurozio and Sondow (2020, The American Mathematical Monthly127(1), 23-32). The paper also connects the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis to very well-poised generalized hypergeometric series. The confluence of ideas presented reflects a multidisciplinarity that accords with the approach and philosophy of Prasanta Chandra Mahalanobis.

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

In the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities$$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

The Dirichlet problem for rectangle and new identities for elliptic integrals and functions

In the paper, results of comparison of two different methods of exact solution of the Dirichlet problem for rectangle are presented, namely, method of conformal mapping and method of variables’ separation. By means of this procedure normal derivative of Green’s function for rectangular domain was expressed via Jacobian elliptic functions. Under approaching to rectangle’s boundaries these formulas give new representations of the Dirac delta function. Moreover in the framework of suggested ideology a number of identities for the complete elliptic integral of the first kind were obtained. These formulas may be applied to summation of both numerical and functional series; also they may be useful for analytic number theory.

A Rational Approximation for the Complete Elliptic Integral of the First Kind

Let K ( r ) be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for K ( r ) . More precisely, we establish the inequality 2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 ( r ′ ) 2 + 110 r ′ + 21 for r ∈ ( 0 , 1 ) , where r ′ = 1 − r 2 . The lower bound is sharp.