# normal forms

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## 1—10 of 46 matching pages

##### 1: 31.14 General Fuchsian Equation

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###### Normal Form

…##### 2: 30.2 Differential Equations

##### 3: 22.18 Mathematical Applications

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►The special case ${y}^{2}=(1-{x}^{2})(1-{k}^{2}{x}^{2})$ is in

*Jacobian normal form*. For any two points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ on this curve, their*sum*$({x}_{3},{y}_{3})$, always a third point on the curve, is defined by the Jacobi–Abel addition law …##### 4: 22.15 Inverse Functions

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►The integrals (22.15.12)–(22.15.14) can be regarded as

*normal forms*for representing the inverse functions. …can be transformed into normal form by elementary change of variables. …##### 5: 31.2 Differential Equations

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###### §31.2(ii) Normal Form of Heun’s Equation

…##### 6: 36.2 Catastrophes and Canonical Integrals

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###### Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension $K$

… ►###### Normal Forms for Umbilic Catastrophes with Codimension $K=3$

… ►For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975). …##### 7: Bibliography

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Normal forms of functions near degenerate critical points, the Weyl groups ${A}_{k},{D}_{k},{E}_{k}$ and Lagrangian singularities.
Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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Critical points of smooth functions, and their normal forms.
Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
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##### 8: 19.14 Reduction of General Elliptic Integrals

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►The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial.
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##### 9: 36.10 Differential Equations

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►In terms of the normal form (36.2.1) the ${\mathrm{\Psi}}_{K}\left(\mathbf{x}\right)$ satisfy the operator equation
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►In terms of the normal forms (36.2.2) and (36.2.3), the ${\mathrm{\Psi}}^{(\mathrm{U})}\left(\mathbf{x}\right)$ satisfy the following operator equations
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