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Whipple Formulae
In the theory of special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression, concerning associated Legendre functions. These formulae have been presented previously in terms of a viewpoint aimed at spherical harmonics, now that we view the equations in terms of toroidal coordinates, whole new symmetries of Legendre functions arise. For associated Legendre functions of the first and second kind, :P_^\biggl(\frac\biggr)= \frac and :Q_^\biggl(\frac\biggr)= -i(\pi/2)^\Gamma(-\nu-\mu)(z^2-1)^e^ P_\nu^\mu(z). These expressions are valid for all parameters \nu, \mu, and z. By shifting the complex degree and order in an appropriate fashion, we obtain Whipple formulae for general complex index interchange of general associated Legendre functions of the first and second kind. These are given by : P_^\mu(z)=\frac\biggl \pi\sin\mu\pi P_^\nu\biggl(\frac\biggr)+\cos\pi(\nu+\mu)e^Q_^\nu\biggl(\frac\biggr) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Functions
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles. Legendre's differential equation The general Legendre equation reads \left(1 - x^2\right) y'' - 2xy' + \left lambda(\lambda+1) - \frac\righty = 0, where the numbers and may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when is an integer (denoted ), and are the Legendre polynomials ; and when is an integer (denoted ), and is also an integer with are the associated Legendre polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis John Welsh Whipple
Francis John Welsh Whipple ScD FInstP (17 March 1876 – 25 September 1943) was an English mathematician, meteorologist and seismologist. From 1925 to 1939, he was superintendent of the Kew Observatory. Biography Whipple was the son of Kew Observatory employees George Mathews Whipple and Elizabeth Beckley, an astronomical photographer. Whipple attended Willington Preparatory School in Putney, where in 1888 he won a scholarship at Merchant Taylors' School. From here, he obtained a scholarship at Trinity College, Cambridge in 1895, where he was placed Second Wrangler in the Mathematical Tripos of 1897. In 1899, he showed that bicycles could be self stable. Career From 1899–1912, he was an assistant master at Merchant Taylors' School, and then worked at the Meteorological Office from 1912. From 1925, he was Assistant-Director of the Meteorological Office and Superintendent of Kew Observatory, ehere he succeeded Charles Chree. Whipple remained in this post until he retired in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Legendre Functions
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles. Legendre's differential equation The general Legendre equation reads \left(1 - x^2\right) y'' - 2xy' + \left lambda(\lambda+1) - \frac\righty = 0, where the numbers and may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when is an integer (denoted ), and are the Legendre polynomials ; and when is an integer (denoted ), and is also an integer with are the associated Legendre polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Coordinates
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci F_1 and F_2 in bipolar coordinates become a ring of radius a in the xy plane of the toroidal coordinate system; the z-axis is the axis of rotation. The focal ring is also known as the reference circle. Definition The most common definition of toroidal coordinates (\tau, \sigma, \phi) is : x = a \ \frac \cos \phi : y = a \ \frac \sin \phi : z = a \ \frac together with \mathrm(\sigma)=\mathrm(z). The \sigma coordinate of a point P equals the angle F_ P F_ and the \tau coordinate equals the natural logarithm of the ratio of the distances d_ and d_ to opposite sides of the focal ring : \tau = \ln \frac. The coordinate ranges are -\pi<\sigma\le\pi, and Coordinate surfaces Surfaces of con ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astronomische Nachrichten
''Astronomische Nachrichten'' (''Astronomical Notes''), one of the first international journals in the field of astronomy, was established in 1821 by the German astronomer Heinrich Christian Schumacher. It claims to be the oldest astronomical journal in the world that is still being published. The publication today specializes in articles on solar physics, extragalactic astronomy, cosmology, geophysics, and instrumentation for these fields. All articles are subject to peer review. Early history The journal was founded in 1821 by Heinrich Christian Schumacher,''Publications of the Astronomical Society of the Pacific'', page 60, v.7 (1895) under the patronage of Christian VIII of Denmark, and quickly became the world's leading professional publication for the field of astronomy. Schumacher edited the journal at the Altona Observatory, then under the administration of Denmark, later part of Prussia, and today part of the German city of Hamburg. Schumacher edited the first 31 issue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
