Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S 2 (for conductivity) and T 2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S 2 and T 2 for different lattices are established. In particular, the value S 2 = π for the square and hexagonal arrays is discussed and T 2 = π/2 for the hexagonal is deduced.

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“…In the sequel, we will use such values for small r and the corresponding singular values of the elliptic integral K(k r ) (see [24,25]), namely …”

Section: Exact Relations Between Lattice Sumsmentioning

confidence: 99%

Closed-form evaluation of two-dimensional static lattice sums

2016

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“…As further justification of our elliptic approach, we draw attention to the fact that the internal energy on the arbitrary odd sphere, S d , is a polynomial in just two quantities, see (18) and (44). For scalars this is a simple consequence of standard analytical properties of the Weierstrass function, ℘, but seems not so obvious in the Epstein formulation.…”

Section: Conclusion and Extensionsmentioning

confidence: 99%

Elliptic functions and temperature inversion symmetry on spheres

Dowker

Kirsten

2002

Nuclear Physics B

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Finite temperature boson and fermion field theories on the space-time manifolds R×S d are discussed with one eye on the questions of temperature inversion symmetry and modular invariance. For conformally invariant theories it is shown that the total energy at any temperature for any dimension, d, is given as a power series in the d = 3 and d = 5 energies, for scalars, and the d = 1 and d = 3 energies for spinors. Further, these energies can be given in finite terms at specific temperatures associated with singular moduli of elliptic function theory. Some examples are listed and numbers given.

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“…We note also that for Γ (n/24) with n integer, elliptic integral algorithms are known that converge as fast as those for π [27], [22].…”

Section: It Turns Out That For Any Rationals R Smentioning

confidence: 99%