A high accurate method for solving an inverse problem of the Laplace equation in detection of a Robin coefficient

This study This work deals with an inverse problem for the harmonic equation to recover a Robin coefficient on a non-accessible part of a circle from Cauchy data measured on an accessible part of that circle. By assuming that the available data has a Fourier expansion, we adopt the Modified Collocation Trefftz Method (MCTM) to solve this problem. We use the truncation regularization method in combination with the collocation technique to approximate the solution, and the conjugate gradient method to obtain the coefficients, thus completing the missing Cauchy data. We recommend the least squares method to achieve a better stability. Finally, we illustrate the feasibility of this method with numerical examples.

Asymptotic Approximation of the Apostol-Tangent Polynomials Using Fourier Series

Asymptotic approximations of the Apostol-tangent numbers and polynomials were established for non-zero complex values of the parameter λ. Fourier expansion of the Apostol-tangent polynomials was used to obtain the asymptotic approximations. The asymptotic formulas for the cases λ=1 and λ=−1 were explicitly considered to obtain asymptotic approximations of the corresponding tangent numbers and polynomials.

Rationality of the Petersson Inner Product of Cohen’s Kernels

By explicitly calculating and then analytically continuing the Fourier expansion of the twisted double Eisenstein series [Formula: see text] of Diamantis and O’Sullivan, we prove a formula of the Petersson inner product of Cohen’s kernel and one of its twists, and obtain a rationality result. This extends a result of Kohnen and Zagier.

A few remarks on the values of the Bernoulli polynomials at rational arguments and some relations with ζ(2k + 1)

A problem posed by Lehmer in 1938 asks for simple closed formulae for the values of the even Bernoulli polynomials at rational arguments in terms of the Bernoulli numbers. We discuss this problem based on the Fourier expansion of the Bernoulli polynomials. We also give some necessary and sufficient conditions for ζ(2k + 1) be a rational multiple of π2k+1.

A generalized Davenport expansion

We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.

Counting quantum states and field modes

“Counting quantum states and field modes” deals with wave modes identified as terms in a Fourier expansion made within a large arbitrary volume. Travelling-wave modes are preferred as they are eigenstates of momentum; counting modes is also made straightforward, whence the density of states. This is in contrast to a fashion that works instead with standing waves.

Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,infty). In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.