See [[Weierstrass elliptic functions|Weierstrass elliptic functions]].
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See [[Weierstrass elliptic functions|Weierstrass elliptic functions]].
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...z)$ a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral is an integral of the form
...arily, that is, by algebraic functions in $z$ or in the logarithms of such functions. For example,
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$#C+1 = 11 : ~/encyclopedia/old_files/data/A012/A.0102160 Amplitude of an elliptic integral
in an [[Elliptic integral|elliptic integral]] of the first kind
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#REDIRECT [[Weierstrass elliptic functions]]
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...ingle-valued composite functions of the form $f(u(z))$ in the case of an [[elliptic integral]]
...owed that its solution led to new transcendental elliptic functions (cf. [[Elliptic function]]).
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$#C+1 = 8 : ~/encyclopedia/old_files/data/L058/L.0508120 Lemniscate functions
...Elliptic function|Elliptic function]]). They arise in the inversion of the elliptic integral of special form
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...lliptic integral]] in Legendre normal form. For example, in the incomplete elliptic integral of the first kind,
...on of the [[Jacobi elliptic functions]], which arise from the inversion of elliptic integrals of the form \eqref{*}.
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...hyper-elliptic curve is $g =(n-1)/2$, so that, for various odd $n$, hyper-elliptic curves are birationally inequivalent.
...zation of hyper-elliptic curves. A further characterization is that hyper-elliptic curves have exactly $2g+2$ [[Weierstrass point]]s.
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One of the three fundamental [[Jacobi elliptic functions|Jacobi elliptic functions]]. It is denoted by
...ed as follows in terms of the Weierstrass sigma-function, the Jacobi theta-functions or a series:
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one obtains elliptic integrals (cf. [[Elliptic integral|Elliptic integral]]), while the cases $ m = 5, 6 $
are sometimes denoted as ultra-elliptic.
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''elliptic cosine''
One of the three basic [[Jacobi elliptic functions|Jacobi elliptic functions]], denoted by
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''elliptic sine''
One of the three basic [[Jacobi elliptic functions|Jacobi elliptic functions]], written as
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...cal methods]]; [[Elliptic partial differential equation, numerical methods|Elliptic partial differential equation, numerical methods]]; [[Differential equation
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...ure in the boundary conditions of boundary value problems for second-order elliptic equations. The problem is then called a problem with oblique derivative. Se
...ection field $l$ on $S$ has the form $l=(l_1,\ldots,l_n)$, where $l_i$ are functions of the points $P\in S$ such that $\sum_{i=1}^n(l_i)^2=1$, then the oblique
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$#C+1 = 64 : ~/encyclopedia/old_files/data/E035/E.0305470 Elliptic function
...dic function]] that is meromorphic in the finite complex $ z $-plane. An elliptic function has the following basic properties.
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$#C+1 = 102 : ~/encyclopedia/old_files/data/E035/E.0305490 Elliptic integral
...on is possible, then (1) is said to be a [[Pseudo-elliptic integral|pseudo-elliptic integral]].
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A Tate curve is a uniformization of an [[elliptic curve]] having stable bad reduction with the help of a $q$-parametrization.
...l $j$-invariant). In the case of stable bad reduction one can construct an elliptic curve $E_q$ over $K$, which analytically is $K^*/q^{\mathbb{Z}}$ (where $q^
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are integers. Analytic functions of one complex variable with more than two primitive periods do not exist,
...n]]). The generalization of the concept of an elliptic function to include functions $ f ( z _ {1} \dots z _ {n} ) $
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for all smooth functions $ \phi $(
...ons (cf. [[Strong solution|Strong solution]])? For example, in the case of elliptic equations, every weak solution is strong.
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