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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] |
Weber's Modular Function
In mathematics, the Weber modular functions are a family of three functions ''f'', ''f''1, and ''f''2,''f'', ''f''1 and ''f''2 are not modular functions (per the Wikipedia definition), but every modular function is a rational function in ''f'', ''f''1 and ''f''2. Some authors use a non-equivalent definition of "modular functions". studied by Heinrich Martin Weber. Definition Let q = e^ where ''τ'' is an element of the upper half-plane. Then the Weber functions are :\begin \mathfrak(\tau) &= q^\prod_(1+q^) = \frac = e^\frac,\\ \mathfrak_1(\tau) &= q^\prod_(1-q^) = \frac,\\ \mathfrak_2(\tau) &= \sqrt2\, q^\prod_(1+q^)= \frac. \end These are also the definitions in Duke's paper ''"Continued Fractions and Modular Functions"''.https://www.math.ucla.edu/~wdduke/preprints/bams4.pdf ''Continued Fractions and Modular Functions'', W. Duke, pp 22-23 The function \eta(\tau) is the Dedekind eta function and (e^)^ should be interpreted as e^. The descriptions as \eta quotients immediately im ... [...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] |
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] |
