Related papers: Fun With Fourier Series
We apply a seemingly forgotten series expression of N\"orlund for the psi function to express infinite series involving inverse factorials in closed form. Many of such series contain products of Catalan numbers and (odd) harmonic numbers.…
Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum…
In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the corresponding transformation that…
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
We present a new form of the Machin-like formula for $\pi$ that can be generated by using iteration. This form of the Machin-like formula may be promising for computation of the constant $\pi$ due to rapidly increasing integers at each step…
Kummer's Fourier series for the log gamma function is well known, having been discovered in 1847. In this paper we develop a corresponding Fourier series for the logarithm of the Barnes double gamma function (and the method may be easily…
The probability density function of the random flight with isotropic initial conditions is obtained by an expansion in the number of collisions and the in the spatial harmonics of the solution, as in a Fourier series. The method holds for…
Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has…
We introduce a shifted version of the binomial theorem, and use it to study some remarkable trigonometric integrals and their explicit rewriting in terms of binomial multiple sums. Motivated by the expressions of area generating functions…
This chapter presents an overview of techniques used for the analysis, edition, and synthesis of time series, with a particular emphasis on motion data. The use of mixture models allows the decomposition of time signals as a superposition…
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers…
Recently we have reported a new method of rational approximation of the sinc function obtained by sampling and the Fourier transforms. However, this method requires a trigonometric multiplier that originates from shifting property of the…
In this paper, we investigate the Ces\'aro means of Fourier series with respect to general orthonormal systems (ONS), when the function \( f \) belongs to a certain differentiable class of functions. It is well known that the membership of…
This work introduces a new functional series for expanding an analytic function in terms of an arbitrary analytic function. It is generally applicable and straightforward to use. It is also suitable for approximating the behavior of a…
A number of elegant approaches have been developed for the identification of quantum circuits which can be efficiently simulated on a classical computer. Recently, these methods have been employed to demonstrate the classical simulability…
In this work, we describe examples for calculating the 1-D circular convolution of signals represented by 3-qubit superpositions. The case is considered, when the discrete Fourier transform of one of the signals is known and calculated in…
We study the convergence sets of a class of alternating series. Among other things, our results establish the convergence of the series $\sum_n (-1)^n|\sin n|/n$.
The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves.…
Borwein integrals are one of the most popularly known phenomena in contemporary mathematics. They were found in 2001 by David Borwein and Jonathan Borwein and consist of a simple family of integrals involving the cardinal sine function…
In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of $\sinc$ functions. We then give a general formula to compute the integral on the real line of the…