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Neville Theta Function
In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows: : \theta_c(z,m)=\frac \,\, \sum _^\infty (q(m))^ \cos \left(\frac \right) : \theta_d(z,m)=\frac\,\,\left( 1+2\,\sum _^\infty (q(m))^ \cos \left( \frac \right) \right) : \theta_n(z, m) =\frac \,\,\left( 1+2\sum _^\infty (-1)^k (q(m))^ \cos \left(\frac \right) \right) : \theta_s(z, m)=\frac\,\, \sum_^\infty (-1)^k (q(m))^ \sin\left(\frac \right) where: ''K(m)'' is the complete elliptic integral of the first kind, K'(m)=K(1-m), and q(m)=e^ is the elliptic nome. Note that the functions ''θp(z,m)'' are sometimes defined in terms of the nome ''q(m)'' and written ''θp(z,q)'' (e.g. NIST). The functions may also be written in terms of the ''τ'' parameter ''θp(z, τ)'' where q=e^. Relationship to other functions The Neville theta functions may be expressed in terms of the Jacobi theta functions :\theta_s(z, \tau)=\theta_3^2(0, \tau)\theta_1(z', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Eric Harold Neville
Eric Harold Neville, known as E. H. Neville (1 January 1889 London, England – 22 August 1961 Reading, Berkshire, England) was an English mathematician. A heavily fictionalised portrayal of his life is rendered in the 2007 novel '' The Indian Clerk''. He is the one who convinced Srinivasa Ramanujan to come to England. Early life and education Eric Harold Neville was born in London on 1 January 1889. He attended the William Ellis School, where his mathematical abilities were recognised and encouraged by his mathematics teacher, T. P. Nunn. In 1907, he entered Trinity College, Cambridge. He graduated second wrangler two years later. He was elected to a Fellowship at Trinity College. While there he became acquainted with other Cambridge fellows, most notably Bertrand Russell and G. H. Hardy. In 1913 Neville married Alice Farnfield (1875-1956); they had a son Eric Russell Neville in 1914 who died before his first birthday. Neville remained married to Alice until her death. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Jacobi Elliptic Functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacobi elliptic functions denoted by \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |