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1: 3.6 Linear Difference Equations
It therefore remains to apply a normalizing factor Λ . … The normalizing factor Λ can be the true value of w 0 divided by its trial value, or Λ can be chosen to satisfy a known property of the wanted solution of the form … …
2: 3.7 Ordinary Differential Equations
The eigenvalues λ k are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy …
3: 18.2 General Orthogonal Polynomials
The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials p n ( x ) uniquely up to constant factors, which may be fixed by suitable normalization. …
4: 28.5 Second Solutions fe n , ge n
The factors C n ( q ) and S n ( q ) in (28.5.1) and (28.5.2) are normalized so that
28.5.5 ( C n ( q ) ) 2 0 2 π ( f n ( x , q ) ) 2 d x = ( S n ( q ) ) 2 0 2 π ( g n ( x , q ) ) 2 d x = π .
5: 33.13 Complex Variable and Parameters
33.13.2 R = ( 2 + 1 ) C ( η ) / C - 1 ( η ) .
6: Bibliography C
  • B. C. Carlson and J. FitzSimons (2000) Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.
  • B. C. Carlson (1964) Normal elliptic integrals of the first and second kinds. Duke Math. J. 31 (3), pp. 405–419.
  • D. C. Cronemeyer (1991) Demagnetization factors for general ellipsoids. J. Appl. Phys. 70 (6), pp. 2911–2914.
  • Cunningham Project (website)
  • S. W. Cunningham (1969) Algorithm AS 24: From normal integral to deviate. Appl. Statist. 18 (3), pp. 290–293.
  • 7: 19.16 Definitions
    19.16.1 R F ( x , y , z ) = 1 2 0 d t s ( t ) ,
    19.16.2 R J ( x , y , z , p ) = 3 2 0 d t s ( t ) ( t + p ) ,
    19.16.2_5 R G ( x , y , z ) = 1 4 0 1 s ( t ) ( x t + x + y t + y + z t + z ) t d t .
    19.16.4 s ( t ) = t + x t + y t + z .
    It should be noted that the integrals (19.16.1)–(19.16.2_5) have been normalized so that R F ( 1 , 1 , 1 ) = R J ( 1 , 1 , 1 , 1 ) = R G ( 1 , 1 , 1 ) = 1 . …
    8: 28.12 Definitions and Basic Properties
    In consequence, for the Floquet solutions w ( z ) the factor e π i ν in (28.2.14) is no longer ± 1 . …
    28.12.2 λ ν ( - q ) = λ ν ( q ) = λ - ν ( q ) .
    As in §28.7 values of q for which (28.2.16) has simple roots λ are called normal values with respect to ν . For real values of ν and q all the λ ν ( q ) are real, and q is normal. … If q is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of z and q by the normalization
    9: 10.22 Integrals
    10.22.72 0 J μ ( a t ) J ν ( b t ) J ν ( c t ) t 1 - μ d t = ( b c ) μ - 1 sin ( ( μ - ν ) π ) ( sinh χ ) μ - 1 2 ( 1 2 π 3 ) 1 2 a μ e ( μ - 1 2 ) i π Q ν - 1 2 1 2 - μ ( cosh χ ) , μ > - 1 2 , ν > - 1 , a > b + c , cosh χ = ( a 2 - b 2 - c 2 ) / ( 2 b c ) .
    10: Bibliography
  • A. G. Adams (1969) Algorithm 39: Areas under the normal curve. The Computer Journal 12 (2), pp. 197–198.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • 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).