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11: Bibliography L
  • G. Labahn and M. Mutrie (1997) Reduction of Elliptic Integrals to Legendre Normal Form. Technical report Technical Report 97-21, Department of Computer Science, University of Waterloo, Waterloo, Ontario.
  • 12: 3.6 Linear Difference Equations
    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 formLet us assume the normalizing condition is of the form w 0 = λ , where λ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns w 1 ( N ) , w 2 ( N ) , , w N - 1 ( N ) ; see §3.2(ii). … For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …
    13: 31.15 Stieltjes Polynomials
    The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space L ρ 2 ( Q ) . …
    14: 33.2 Definitions and Basic Properties
    The normalizing constant C ( η ) is always positive, and has the alternative form
    15: 8.23 Statistical Applications
    Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). The function B x ( a , b ) and its normalization I x ( a , b ) play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …
    16: 30.15 Signal Analysis
    The sequence ϕ n , n = 0 , 1 , 2 , forms an orthonormal basis in the space of σ -bandlimited functions, and, after normalization, an orthonormal basis in L 2 ( - τ , τ ) . …
    17: 33.5 Limiting Forms for Small ρ , Small | η | , or Large
    §33.5 Limiting Forms for Small ρ , Small | η | , or Large
    §33.5(i) Small ρ
    33.5.6 C ( 0 ) = 2 ! ( 2 + 1 ) ! = 1 ( 2 + 1 ) !! .
    §33.5(iii) Small | η |
    §33.5(iv) Large
    18: 28.14 Fourier Series
    28.14.1 me ν ( z , q ) = m = - c 2 m ν ( q ) e i ( ν + 2 m ) z ,
    and the normalization relation
    28.14.5 m = - ( c 2 m ν ( q ) ) 2 = 1 ;
    19: 19.31 Probability Distributions
    R G ( x , y , z ) and R F ( x , y , z ) occur as the expectation values, relative to a normal probability distribution in 2 or 3 , of the square root or reciprocal square root of a quadratic form. …
    20: 28.12 Definitions and Basic Properties
    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