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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The ratio of the two basic [[Jacobi elliptic functions]]:
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...serve as elliptic functions which generate the algebraic field of elliptic functions with given primitive periods.
...) $ and $ \wp ^ \prime (z) $ generate the algebraic field of elliptic functions with given periods. The simply-periodic trigonometric function which serves
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which is interpreted in the sense of the theory of generalized functions, where $ \delta $
...fficients, and also for arbitrary elliptic equations. For example, for the elliptic equation
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...rtain elliptic boundary value problem, in terms of the coefficients of the elliptic equation and of the boundary data. Let
be a uniformly elliptic operator in a region $ \Omega _{1} $
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that is, the representation of all possible rational functions of $ w _ {1} $,
...this leads to doubly-periodic elliptic functions (cf. [[Elliptic function|Elliptic function]]). For example, the inversion of an integral of the first kind in
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$#C+1 = 161 : ~/encyclopedia/old_files/data/J054/J.0504050 Jacobi elliptic functions
...m, by N.H. Abel. Jacobi's construction is based on an application of theta-functions (cf. [[Theta-function|Theta-function]]).
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...tic function|Analytic function]]; [[Elliptic partial differential equation|Elliptic partial differential equation]]). To apply the transform, complex (independ
(where $J_0$ is a Bessel function, cf. [[Bessel functions|Bessel functions]]).
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...ends linearly on the parameters in the following way. Let $F_1,F_2,F_3$ be functions of two variables, no one of which is a linear combination of the other two.
...gh a single point. If this point is a finite point, then the net is called elliptic, if it is a point at infinity, then the net is called parabolic.
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''elliptic modular function, of one complex variable''
...e of the general theory of automorphic functions. In the theory of modular functions the following [[Theta-series|theta-series]] are used as basic modular forms
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...a number of cases, for example, for an ordinary differential operator, for elliptic operators and for differential operators with constant coefficients.
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...oids can be used to classify the points on a surface (see [[Elliptic point|Elliptic point]]; [[Hyperbolic point|Hyperbolic point]]; [[Parabolic point|Parabolic
...tives up to and including order 2 of the difference $p(x,y)-s(x,y)$ of the functions $p(x,y)$ and $s(x,y)$ describing the paraboloid and the surface are all zer
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...n by zero remain smooth up to the boundary are taken by these operators to functions that are again smooth up to the boundary. Here the extension by zero is car
...-differential operator has an asymptotic expansion in positive homogeneous functions $ a _ \alpha ( x, \xi ) $ (where $ \alpha $
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The functions $ \phi _ {k} $,
For a linear uniformly-elliptic equation
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...bolic differential equations (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]; [[Parabolic partial differential equation|
continuous functions with $ K _ {1} ( x,y ) > 0 $,
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...etric functions]] or [[Weierstrass elliptic functions|Weierstrass elliptic functions]] and are automorphic; their group is the group of motions of the Euclidean
...up and regular polygons, such Schwarz functions are also called polyhedral functions.
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A series of functions used in the representation of automorphic forms and functions (cf. [[Automorphic form|Automorphic form]]; [[Automorphic function|Automorp
...ped the theory of theta-series in connection with the study of automorphic functions of one complex variable. Let $ \Gamma $
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in which the real-valued functions $ a _ {ij} ( x) $,
equation (1) is called elliptic at the point $ x _ {0} $;
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...n-characteristic interval of parabolic degeneracy. The Tricomi equation is elliptic for $y>0$, hyperbolic for $y<0$ and degenerates to an equation of parabolic
...eam function of a plane-parallel steady-state gas flow, $k(y)$ and $y$ are functions of the velocity of the flow, which are positive at subsonic and negative at
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...nometric functions]] can be described as the class of meromorphic periodic functions with period $ 2 \pi $
...D valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''2''' , Chelsea (1977) (Translated from Russia
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In the broad sense, a set of several classes of functions that arise in the solution of both theoretical and applied problems in vari
In the narrow sense, the special functions of mathematical physics, which arise when solving partial differential equa
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A problem involving partial differential equations in which the functions specifying the initial (boundary) conditions are not continuous.
In this case the discontinuities of the initial functions $ \phi _ {0} $
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has a normal, and let the following second-order elliptic equation be given:
are continuous functions defined on $ \Gamma $.
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...the complementary modulus (in [[Jacobi elliptic functions|Jacobi elliptic functions]]) or the modulus of a [[Congruence|congruence]]. Cf. also [[Norm on a fiel
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...–Jacobi and Eisenstein–Weierstrass theory of elliptic and elliptic modular functions between 1820– 1850 (see [[#References|[a2]]]). The function $ k ( 2 ) $
...a function of the quotient of the periods of a certain [[Elliptic integral|elliptic integral]]. It corresponds to the level-two congruence subgroup of the [[Mo
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...xtension of it). The rational functions on a modular curve lift to modular functions (of a higher level) and form a field; the automorphisms of this field have
...es|[8]]]). In particular, there is a hypothesis that each [[Elliptic curve|elliptic curve]] over $ \mathbf Q $
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...eteq D$. In these definitions, $L_1$ can be replaced by the class $L_p$ of functions whose $p$-th powers are locally integrable. The class most often used is $L
In the case of an elliptic equation \eqref{*} both notions of a strong solution coincide.
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...liptic differential operator (cf. [[Elliptic partial differential equation|Elliptic partial differential equation]]) of order $ m $,
With every operator (1) there is associated the homogeneous elliptic operator
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Let a second-order elliptic partial differential equation be given in $ \mathbf R ^ {n} $,
are sufficiently-smooth functions in $ D $
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...postulates a deep connection between elliptic curves (cf. [[Elliptic curve|Elliptic curve]]) over the rational numbers and modular forms (cf. [[Modular form|Mo
...for $L(E,s)$. It is prototypical of a general relationship between the $L$-functions attached to arithmetic objects and those attached to automorphic forms (cf.
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...epresenting the two-dimensional analogue of a [[Hyper-elliptic curve|hyper-elliptic curve]]. A non-singular algebraic projective surface $ X $
is said to be a double plane if its field of rational functions $ k ( X ) $
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