Error bounds and exponential improvement for Hermite's asymptotic expansion for the Gamma function

Nemes, Gergýo

doi:10.2298/aadm130124002n

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…We need the asymptotics of the Gamma function too, whose Stokes phenomena is similar to that of the G-Barnes function above, with the Stokes lines also at ±π/2. One difference is the different decay of the S k (θ) coefficients (Stokes multipliers), where the quadratic decay in the G-Barnes is a linear decay in the Gamma function case [22]. More crucially, there is a sign difference in the respective Stokes multipliers, as we show in what follows.…”

Section: On Exponential Asymptotics and Dualitymentioning

confidence: 67%

“…Therefore, the large z behavior of the special functions is required, involving naturally the whole complex plane, which leads to consideration of Stokes and anti-Stokes lines and the appearance of exponentially small contributions when crossing them. We can analyze the observables in these terms by using the exponential asymptotics of Barnes and Gamma functions [21,22,23]. Therefore, this theory is suitable to further explore ideas of resurgence and resummation, which have become a subject of considerable interest in the study of gauge theories in recent years (see [23,24], for example, [25,26,27] for supersymmetric gauge theories and localization and [28] for a review).…”

Section: Introductionmentioning

confidence: 99%

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Wilson loops and free energies in 3D $\mathcal{N}=4$ SYM: exact results, exponential asymptotics, and duality

Tierz

2019

Progress of Theoretical and Experimental Physics

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We show that U (N ) 3d N = 4 supersymmetric gauge theories on S 3 with N f massive fundamental hypermultiplets and with a Fayet-Iliopoulos (FI) term are solvable in terms of generalized Selberg integrals. Finite N expressions for the partition function and Wilson loop in arbitrary representations are given. We obtain explicit analytical expressions for Wilson loops with symmetric, antisymmetric, rectangular and hook representations, in terms of Gamma functions of complex argument. The free energy for orthogonal and symplectic gauge group is also given. The asymptotic expansion of the free energy is also presented, including a discussion of the appearance of exponentially small contributions. Duality checks of the analytical expressions for the partition functions are also carried out explicitly.

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Section: On Exponential Asymptotics and Dualitymentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Wilson loops and free energies in 3D $\mathcal{N}=4$ SYM: exact results, exponential asymptotics, and duality

Tierz

2019

Progress of Theoretical and Experimental Physics

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“…Hermite and Barnes gave the asymptotics of log Γ(z + a) as |z| → ∞ when 0 a 1. These shifted argument results and further improvements are described in [Nem13]. See also [Olv74,Ex.…”

Section: Linear Independencementioning

confidence: 69%

A generalization of the Riemann-Siegel formula

O’Sullivan¹

2018

Preprint

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The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable t. Siegel anticipated that this formula could be generalized to include the Hardy-Littlewood approximate functional equation, valid in any vertical strip. We give this generalization for the first time. The asymptotics contain Mordell integrals and an interesting new family of polynomials.

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“…Up to some constant this is precisely the non-perturbative term of the log Γ-function on the Stokes line, cf., [20,21]. The underlying important relation is the transformation (3.11) under varying the moduli t and .…”

Section: Beyond Conifold Pointsmentioning

confidence: 99%

Non-perturbative quantum geometry III

Krefl

2016

J. High Energ. Phys.

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The Nekrasov-Shatashvili limit of the refined topological string on toric CalabiYau manifolds and the resulting quantum geometry is studied from a non-perturbative perspective. The quantum differential and thus the quantum periods exhibit Stokes phenomena over the combined string coupling and quantized Kähler moduli space. We outline that the underlying formalism of exact quantization is generally applicable to points in moduli space featuring massless hypermultiplets, leading to non-perturbative band splitting. Our prime example is local P 1 × P 1 near a conifold point in moduli space. In particular, we will present numerical evidence that in a Stokes chamber of interest the string based quantum geometry reproduces the non-perturbative corrections for the Nekrasov-Shatashvili limit of 4d supersymmetric SU(2) gauge theory at strong coupling found in the previous part of this series. A preliminary discussion of local P 2 near the conifold point in moduli space is also provided.

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Error bounds and exponential improvement for Hermite's asymptotic expansion for the Gamma function

Cited by 7 publications

References 21 publications

Wilson loops and free energies in 3D $\mathcal{N}=4$ SYM: exact results, exponential asymptotics, and duality

Wilson loops and free energies in 3D $\mathcal{N}=4$ SYM: exact results, exponential asymptotics, and duality

A generalization of the Riemann-Siegel formula

Non-perturbative quantum geometry III

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