Results 161 to 170 of about 1,712 (185)
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Elliptic integrals, the forgotten functions
European Journal of Physics, 2001Summary: Ten simple and useful complete elliptic integrals are presented. Their application is illustrated in elementary examples from electromagnetism. Some background is provided involving their history and their relationship to elliptic functions.
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Improper Integrals. Elliptic Integrals and Functions
1983When f is R-integrable over [a, b] then its indefinite integral F, defined as $$F\left( x \right) = \int_a^x {f\left( t \right)dt\,\,\,\,for\,\,\,\,x \in \left[ {a,b} \right]} ,$$ (1.1) is continuous on [a,b] (Theorem XIII.6.3). Hence, $$_{x \to b - }^{\lim }\int_a^x {f\left( t \right)dt = \int_a^b {f\left( t \right)dt.} }$$ (1.2)
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Integral involving elliptic function
Rendiconti del Circolo Matematico di Palermo, 1967It is shown here that an IntegralI of the form\(\int {\frac{{dx}}{{[Ax^0 + {\rm B}x^4 + Cx^2 + D]^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} }}} \), whereA, B, C, D are real constants, can, under certain conditions, be expressed as an Elliptic Integral of the first kind.
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New integral identity for elliptic functions
Soviet Journal of Quantum Electronics, 1991Analytic solutions of what are known as the reduced equations for the amplitudes and phases of oscillations and waves in nonlinear systems are used to establish a new integral identity for elliptic functions. The consequence of this identity is a new relationship between incomplete elliptic integrals of the first and third kinds.
Valentin G Dmitriev, D A Guk
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Integral differentials of elliptic function fields
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 2004Let \(K\) be an arithmetic function field of transcendence degree \(r\) over a number field \(k\) and denote by \(A\) the ring of integers of \(k\). In his fundamental work ``Geometria aritmetica'' published in [Ann. Mat. Pura Appl., IV. Ser. 45 (1958; Zbl 0142.18101)], which laid the foundations for an algebraic differential calculus on arithmetic ...
Kunz, E., Waldi, R.
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Complex Functions and Elliptic Integrals
2015This chapter considers how elliptic functions and complex functions were first brought together. This was an important step for both subjects, which, as Jacobi noted in his lectures, seemed to be kept apart by the complications resulting from the two-valued nature of the integrand in the elliptic integrals.
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Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions
Integral Transforms and Special Functions, 2017Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the indep...
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Precise and Fast Computation of Elliptic Integrals and Functions
2015 IEEE 22nd Symposium on Computer Arithmetic, 2015Summarized is the recent progress of the new methods to compute Legendre's complete and incomplete elliptic integrals of all three kinds and Jacobian elliptic functions. Also reviewed are the entirely new methods to (i) compute the inverse functions of complete elliptic integrals, (ii) invert a general incomplete elliptic integral numerically, and (iii)
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Fourier Expansions of Rational Fractions of Elliptic Integrals and Jacobian Elliptic Functions
SIAM Journal on Mathematical Analysis, 1980The Fourier expansions of rational fractions with numerators consisting of various combinations of $sn(u,k),cn(u,k),dn(u,k)$, and the periodic parts of the elliptic integrals $E(am,\, u,k)$ and $\Pi (am\, u,\alpha ^2 ,k)$, and denominators consisting of the first or second powers of $1 \pm \beta cn\,u$ or $1 - \alpha ^2 sn^2 u$ are listed.
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