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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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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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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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Chapter 4 Complex Functions and Elliptic Integrals
2013In this chapter we consider how elliptic function theory and complex variable theory were finally drawn together in the 1830s and 1840s. As the recognition of the importance of the work of Abel and Jacobi grew, mathematicians came to feel that it was unsatisfactory to base the theory of elliptic functions on the inversion of many-valued integrals.
Umberto Bottazzini, Jeremy Gray
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Numerical evaluation of iterated integrals related to elliptic Feynman integrals
Computer Physics Communications, 2021Moritz Walden, Weinzierl Stefan
exaly
Elliptic integrals, elliptic functions and modular forms in quantum field theory
2019This book includes review articles in the field of elliptic integrals, elliptic functions and modular forms intending to foster the discussion between theoretical physicists working on higher loop calculations and mathematicians working in the field of modular forms and functions and analytic solutions of higher order differential and difference ...
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Elliptic Functions and Elliptic Integrals
1997Viktor Prasolov, Yuri Solovyev
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