scholarly journals Oscillatory Asymptotics for the Airy Kernel Determinant on Two Intervals

2020 ◽  
Author(s):  
Elliot Blackstone ◽  
Christophe Charlier ◽  
Jonatan Lenells
Keyword(s):  
Fredholm Determinant ◽  
Theta Functions ◽  
Explicit Formulas ◽  
Airy Kernel ◽  
Jacobi Theta Functions

Abstract We obtain asymptotics for the Airy kernel Fredholm determinant on two intervals. We give explicit formulas for all the terms up to and including the oscillations of order $1$, which are expressed in terms of Jacobi $\theta $-functions.

scholarly journals Second-order integrable Lagrangians and WDVV equations

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
E. V. Ferapontov ◽  
M. V. Pavlov ◽  
Lingling Xue
Keyword(s):  
Lagrangian Density ◽  
Theta Functions ◽  
Second Order ◽  
Elementary Functions ◽  
Lagrange Equations ◽  
Wdvv Equations ◽  
Integrability Conditions ◽  
Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

FINITE EUCLIDEAN MAGNETIC GROUP AND THETA FUNCTIONS

1992 ◽  
Vol 07 (19) ◽  
pp. 4671-4691 ◽  
Author(s):  
S. FUBINI
Keyword(s):  
Magnetic Field ◽  
Phase Transitions ◽  
Constant Magnetic Field ◽  
Theta Functions ◽  
Type Ii ◽  
Finite Rotations ◽  
Type Ii Superconductors ◽  
Modular Transformations ◽  
Jacobi Theta Functions

The Euclidean magnetic group of translations and rotations in a constant magnetic field is discussed in detail. The eigenfunctions of finite magnetic translations are shown to be related to the quasi periodic Jacobi theta functions, whose group theoretical properties under modular transformations are simply discussed. Invariance under finite rotations is very important; it leads to the two fundamental lattices of 60° and 90° already appearing in the theory of the phase transitions of Type II superconductors.

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