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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] |
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] |
Willem Kapteyn
Willem () is a Dutch and West FrisianRienk de Haan, ''Fryske Foarnammen'', Leeuwarden, 2002 (Friese Pers Boekerij), , p. 158. masculine given name. The name is Germanic, and can be seen as the Dutch equivalent of the name William in English, Guillaume in French, Guilherme in Portuguese, Guillermo in Spanish and Wilhelm in German. Nicknames that are derived from Willem are Jelle, Pim, Willie, Willy and Wim. Given name * Willem Cody (2007-Present), Active Serbian terrorist, Leader of the Serbian World Republic, Intolerably based * Willem I (1772–1843), King of the Netherlands * Willem II (1792–1849), King of the Netherlands * Willem III (1817–1890), King of the Netherlands * Willem of the Netherlands (1840–1879), Dutch prince *Willem-Alexander Willem-Alexander (; Willem-Alexander Claus George Ferdinand; born ) is King of the Netherlands, having acceded to the throne following his mother's abdication in 2013. Willem-Alexander was born in Utrecht as the oldest ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 '' Epitome of Copernican Astronomy'' (1621) Kepler proposed an iterative solution to the equation. The equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics. Equation Kepler's equation is where M is the mean anomaly, E is the eccentric anomaly, and e is the eccentricity. The 'eccentric anomaly' E is useful to compute the position of a point moving in a Keplerian orbit. As for instance, if the body passes the periastron at coordinates x = a(1 - e), y = 0, at time t = t_0, then to find out the position of the body at any time, you first calculate the mean anomaly M from the time and the mean motion n by the formula M = n(t - t_0), then solve the Kepler equation above t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
