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Lommel Function
The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation: : z^2 \frac + z \frac + (z^2 - \nu^2)y = z^. Solutions are given by the Lommel functions ''s''μ,ν(''z'') and ''S''μ,ν(''z''), introduced by , :s_(z) = \frac \left Y_ (z) \! \int_^ \!\! x^ J_(x) \, dx - J_\nu (z) \! \int_^ \!\! x^ Y_(x) \, dx \right :S_(z) = s_(z) + 2^ \Gamma\left(\frac\right) \Gamma\left(\frac\right) \left(\sin \left \mu - \nu)\frac\rightJ_\nu(z) - \cos \left \mu - \nu)\frac\rightY_\nu(z)\right), where ''J''ν(''z'') is a Bessel function of the first kind and ''Y''ν(''z'') a Bessel function of the second kind. See also * Anger function * Lommel polynomial A Lommel polynomial ''R'm'',ν(''z''), introduced by , is a polynomial in 1/''z'' giving the recurrence relation :\displaystyle J_(z) = J_\nu(z)R_(z) - J_(z)R_(z) where ''J''ν(''z'') is a Bessel function of the first kind. They are given ... * Struve function * Weber f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugen Von Lommel
Eugen Cornelius Joseph von Lommel (19 March 1837, Edenkoben – 19 June 1899, Munich) was a German physicist. He is notable for the Lommel polynomial, the Lommel function, the Lommel–Weber function, and the Lommel differential equation. He is also notable as the doctoral advisor of the Nobel Prize winner Johannes Stark. Lommel was born in Edenkoben in the Palatinate, Kingdom of Bavaria. He studied mathematics and physics at the University of Munich between 1854 and 1858. From 1860 to 1865 he is teacher of physics and chemistry at the canton school of Schwyz. From 1865 to 1867 he taught at the high school in Zürich and was simultaneously Privatdozent at the local university as well as at the polytechnic school. From 1867 to 1868, he was appointed professor of physics at the University of Hohenheim. Finally he was appointed to a chair of experimental physics at Erlangen Erlangen (; East Franconian German, East Franconian: ''Erlang'', Bavarian language, Bavarian: ''Erlang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Differential Equation
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 with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
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 with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anger Function
In mathematics, the Anger function, introduced by , is a function defined as : \mathbf_\nu(z)=\frac \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta and is closely related to Bessel functions. The Weber function (also known as Lommel–Weber function), introduced by , is a closely related function defined by : \mathbf_\nu(z)=\frac \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta and is closely related to Bessel functions of the second kind. Relation between Weber and Anger functions The Anger and Weber functions are related by : \begin \sin(\pi \nu)\mathbf_\nu(z) &= \cos(\pi\nu)\mathbf_\nu(z)-\mathbf_(z), \\ -\sin(\pi \nu)\mathbf_\nu(z) &= \cos(\pi\nu)\mathbf_\nu(z)-\mathbf_(z), \end so in particular if ν is not an integer they can be expressed as linear combinations of each other. If ν is an integer then Anger functions Jν are the same as Bessel functions ''J''ν, and Weber functions can be expressed as finite linear combinations of Struve functions. Power series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lommel Polynomial
A Lommel polynomial ''R''''m'',ν(''z''), introduced by , is a polynomial in 1/''z'' giving the recurrence relation :\displaystyle J_(z) = J_\nu(z)R_(z) - J_(z)R_(z) where ''J''ν(''z'') is a Bessel function of the first kind. They are given explicitly by :R_(z) = \sum_^\frac(z/2)^. See also *Lommel function *Neumann polynomial In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions. The first few polynomials are :O_0^(t)=\frac 1 t, :O_1^(t ... References * * *{{citation, first=Eugen von , last=Lommel, title=Zur Theorie der Bessel'schen Functionen , journal =Mathematische Annalen , publisher =Springer , place=Berlin / Heidelberg , volume =4, issue= 1 , year= 1871 , doi =10.1007/BF01443302 , pages =103–116 Polynomials Special functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Struve Function
In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation: : x^2 \frac + x \frac + \left (x^2 - \alpha^2 \right )y = \frac introduced by . The complex number α is the order of the Struve function, and is often an integer. And further defined its second-kind version \mathbf_\alpha(x) as \mathbf_\alpha(x)=\mathbf_\alpha(x)-Y_\alpha(x). The modified Struve functions are equal to , are solutions of the non-homogeneous Bessel's differential equation: : x^2 \frac + x \frac - \left (x^2 + \alpha^2 \right )y = \frac And further defined its second-kind version \mathbf_\alpha(x) as \mathbf_\alpha(x)=\mathbf_\alpha(x)-I_\alpha(x). Definitions Since this is a non-homogeneous equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the Bessel functions, and the particular solution may be chosen as the corresponding Struve fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weber Function (other)
In mathematics, Weber function can refer to several different families of functions, mostly named after the physicist H. F. Weber or the mathematician H. M. Weber: * Weber's modular functions f, f_1, f_2 named after the mathematician H. M. Weber * Weber functions Eν are solutions of an inhomogeneous Bessel equation, and are linear combinations of Anger functions if ν is not an integer, or linear combinations of Struve function In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation: : x^2 \frac + x \frac + \left (x^2 - \alpha^2 \right )y = \frac introduced by . The complex number α is the order of the Struve functio ...s if ν is an integer * Weber–Hermite function is another name for parabolic cylinder functions, which are solutions of Weber's (differential) equation * {{Mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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