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The article focuses on computation and proof of formulas for 2k by mathematician Euler Leonhard. Topics discussed include Bernoulli polynomials definition and properties, using a recurrence formula for evaluation of integers, application of elementary trigonometric identities of cosines and sines for obtaining integral telescoping series and Fourier series being coefficients of Bernoulli polynomials.

The article focuses on the Cauchy problem for the n-dimensional wave equation through the application of elementary properties of the Fourier transform. It discusses various methods used to solve the Cauchy problem for the wave equation in higher dimension including the finite part of a divergent integral, analytic continuation of an integral of fractional order, and non-self-adjoint equations. It mentions that the original solution to the Cauchy problem is in terms of Poisson spherical means.

The article offers the author's insights on how to quantisize boundary value problems with the use of linear dispersion methods. It mentions that the quantisized structures are observed in optics and quantum mechanics which resulted to the investigation on their implication to wave mechanics. Meanwhile, the Fourier series expansions are used to solve initial value problems in the dispersive wave equation.

The article attempts to demonstrate that a variety of trigonometric sums have unexpected closed forms by relating them to cognate integrals. It calculates the sinc sums through the equation sinc(x) := sin (x)/x when x not equal to 0 and sinc(0) := 1. It shows that the theorems for integrals proven in a certain equation imply analogues for sums. It provides the application when sums and integrals agree based on Boas and Pollard using Fourier analysis. It provides the applications to sinc sums and integrals.

The article offers information on a study on edge detection using fourier coefficients. It states that edge detection and the detection of discontinuities are essential in many fields. The problem of identifying the edges present in a function when given the Fourier coefficients of the function is considered in this research. Relative to this, there are numerical methods that project the Fourier coefficients of a function of interest rather than directly approximating the solution. Moreover, the spectral viscosity method, a numerical method which is employed to solve nonlinear partial differential equations (PDEs), is an example of such methods.

The article focuses on the connections that links Fourier series, catalan numbers and random walks on trees. Catalan numbers have many combinatorial interpretations They satisfy the recursion relations. A tree is a connected graph with no loops such that each vertex is an endpoint of only finite edges. Two distinct vertices of the tree are called neighbors if there is edge connecting them. A path in the tree is a finite or infinite sequences of vertices. It is frequently convenient to single out a specific vertex of a tree and declare it to be a root of the tree. There is a natural distance function defined on the set of vertices of a tree. Transient random walks have the property that with probability one an infinite path will visit any particular vertex only finitely many times. There is a simple test for transience of a random walk on an arbitrary tree with a nearest-neighbor probability. The generalized Catalan numbers have numerous combinatorial interpretations. The generalized Catalan numbers can be derived by means of a Pascal-like triangle.

Presents information on a study which analyzed fourier integrals. Details on the Riemann-Lebesgue Lemma; Analysis of various fourier integrals.

Looks at the connection which demonstrates more similarities between power series and Fourier series, and the explanation of some of the differences between the two series. Cooefficients of both series; Ultraspherical polynomials; Introduction of a set of orthogonal polynomials...now called Legendre polynomials; How the connection is given between Fourier and power series.

Discusses an application of the Fourier series to the most significant digit problem. Concept of Benford's logarithmic law of first significant digits; Definition of the Fourier coefficients; Parvesal's formula; Invariance of the log distribution for the first significant digits of products.

Illustrates the Fourier series of a regular polygon in the complex plane. Examples testing the efficiency of the illustration.

The article focuses on the possibility of character analysis using fourier series. It mentions that character analysis using fourier series is possible when mathematical character is considered such as a homomorphism from a given group into the multiplicative group S1. It states that fourier series is a natural tool in harmonic analysis.