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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