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Schlömilch's Series
Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval (0,\pi) in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived the series in 1857. The real-valued function f(x) has the following expansion: :f(x) = a_0 + \sum_^\infty a_n J_0(nx), where :\begin a_0 &= f(0) + \frac \int_0^\pi \int_0^ u f'(u\sin\theta)\ d\theta\ du, \\ a_n &= \frac \int_0^\pi \int_0^ u\cos nu \ f'(u\sin\theta)\ d\theta\ du. \end Examples Some examples of Schlömilch's series are the following: *Null functions in the interval (0,\pi) can be expressed by Schlömilch's Series, 0 = \frac+\sum_^\infty (-1)^n J_0(nx), which cannot be obtained by Fourier Series. This is particularly interesting because the null function is represented by a series expansion in which not all the coefficients are zero. The series converges only when 0 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function Of The First Kind
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. #Spherical Bessel functions, Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oskar Schlömilch
Oskar Xavier Schlömilch (13 April 1823 – 7 February 1901) was a Germans, German mathematician, born in Weimar, working in mathematical analysis. He took a doctorate at the University of Jena in 1842, and became a professor at Dresden University of Technology, Dresden Polytechnic in 1849. He is now known as the eponym of the Polygamma function, Schlömilch function, a kind of Bessel function. He was also an important textbook writer, and editor of the journal '' Zeitschrift für Mathematik und Physik'', of which he was a founder in 1856. He published in 1868 for the first time the Missing square puzzle, dissection paradox, earlier invented by Sam Loyd. In 1862, he was elected a foreign member of the Royal Swedish Academy of Sciences. See also *Glasser's master theorem#A special case: the Cauchy–Schlömilch transformation, Cauchy–Schlömilch transformation *Schlömilch's series *Taylor's theorem#Explicit formulas for the remainder, Schömilch form of the remainder * Cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Course Of Modern Analysis
''A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions'' (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by Edmund T. Whittaker and George N. Watson, first published by Cambridge University Press in 1902. The first edition was Whittaker's alone, but later editions were co-authored with Watson. History Its first, second, third, and the fourth edition were published in 1902, 1915, 1920, and 1927, respectively. Since then, it has continuously been reprinted and is still in print today. A revised, expanded and digitally reset fifth edition, edited by Victor H. Moll, was published in 2021. The book is notable for being the standard reference and textbook for a generation of Cambridge mathematicians including Littlewood and Godfrey H. Hardy. Mary L. Cartwright studied it as preparation for her final hono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John William Strutt, 3rd Baron Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Among many honors, he received the 1904 Nobel Prize in Physics "for his investigations of the densities of the most important gases and for his discovery of argon in connection with these studies." He served as president of the Royal Society from 1905 to 1908 and as chancellor of the University of Cambridge from 1908 to 1919. Rayleigh provided the first theoretical treatment of the elastic scattering of light by particles much smaller than the light's wavelength, a phenomenon now known as "Rayleigh scattering", which notably explains why the sky is blue. He studied and described transverse surface waves in solids, now known as "Rayleigh waves". He contributed extensively to fluid dynamics, with concepts such as the Rayleigh number (a dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niels Nielsen (mathematician)
Niels Nielsen (2 December 1865, in Ørslev – 16 September 1931, in Copenhagen) was a Danish mathematician who specialised in mathematical analysis. Life and work Nielsen was the son of humble peasants and grew up in the western part of the island of Funen. In 1891 he graduated in mathematics from the University of Copenhagen and in 1895 obtained his doctorate. In 1909 he succeeded Julius Petersen as Professor of Mathematics at the University of Copenhagen. His most original works were on special functions, with an important contribution to the theory of the gamma function. In 1917 he suffered from an illness from which he never fully recovered. From this date onward he became interested in the number theory, Bernoulli numbers, Stirling numbers, and the history of mathematics, writing two books on Danish mathematicians of the time period 1528-1908, and two other books on French mathematicians. Selected publications * ''Om en klasse bestemte integraler og nogle derved definer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kapteyn Series
Kapteyn series is a series expansion of analytic functions on a domain in terms of the Bessel function of the first kind. Kapteyn series are named after Willem Kapteyn, who first studied such series in 1893.Kapteyn, W. (1893). Recherches sur les functions de Fourier-Bessel. Ann. Sci. de l’École Norm. Sup., 3, 91-120. Let f be a function analytic on the domain :D_a = \left\ with a0, \Theta_n(z) is defined by : \Theta_n(z) = \frac14\sum_^\frac\left(\frac\right)^ Kapteyn's series are important in physical problems. Among other applications, the solution E of Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was first derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his '' Epi ... M=E-e\sin E can be expressed via a Kapteyn series: : E=M+2\sum_^\infty\fracJ_n(ne). Relation between the Taylor coefficients and the \alpha_n coefficients ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
