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1: 15.11 Riemann’s Differential Equation
§15.11 Riemann’s Differential Equation
The most general form is given by … for arbitrary λ and μ .
2: 9.17 Methods of Computation
For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). …
3: 9.16 Physical Applications
These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …
4: 8.20 Asymptotic Expansions of E p ( z )
§8.20(i) Large z
8.20.1 E p ( z ) = e z z ( k = 0 n 1 ( 1 ) k ( p ) k z k + ( 1 ) n ( p ) n e z z n 1 E n + p ( z ) ) , n = 1 , 2 , 3 , .
Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …
§8.20(ii) Large p
5: 31.11 Expansions in Series of Hypergeometric Functions
§31.11(ii) General Form
6: 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. More generally, let 𝐀 ( = [ a r , s ] ) and 𝐁 ( = [ b r , s ] ) be real positive-definite matrices with n rows and n columns, and let λ 1 , , λ n be the eigenvalues of 𝐀 𝐁 1 . … For (19.31.2) and generalizations see Carlson (1972b).
7: Bibliography G
  • C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.
  • 8: 3.6 Linear Difference Equations
    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). …
    9: 8.21 Generalized Sine and Cosine Integrals
    8.21.18 f ( a , z ) = si ( a , z ) cos z ci ( a , z ) sin z ,
    8.21.19 g ( a , z ) = si ( a , z ) sin z + ci ( a , z ) cos z .
    8.21.22 f ( a , z ) = 0 sin t ( t + z ) 1 a d t ,
    8.21.23 g ( a , z ) = 0 cos t ( t + z ) 1 a d t .
    10: 10.23 Sums
    For the more general form of expansion …