About the Project
NIST

normal form

AdvancedHelp

(0.000 seconds)

1—10 of 46 matching pages

1: 31.14 General Fuchsian Equation
Normal Form
2: 30.2 Differential Equations
The Liouville normal form of equation (30.2.1) is …
3: 22.18 Mathematical Applications
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
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
§31.2(ii) Normal Form of Heun’s Equation
6: 36.2 Catastrophes and Canonical Integrals
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
  • V. I. Arnol’d (1972) 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).
  • V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
  • V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
  • 8: 19.14 Reduction of General Elliptic Integrals
    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. …
    9: 36.10 Differential Equations
    In terms of the normal form (36.2.1) the Ψ K ( x ) satisfy the operator equation … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations …
    10: 36.12 Uniform Approximation of Integrals
    Define a mapping u ( t ; y ) by relating f ( u ; y ) to the normal form (36.2.1) of Φ K ( t ; x ) in the following way: …