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Liouville%E2%80%93Green approximation theorem

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11: 27.6 Divisor Sums
27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
12: 27.7 Lambert Series as Generating Functions
27.7.6 n = 1 λ ( n ) x n 1 x n = n = 1 x n 2 .
13: 30.2 Differential Equations
The Liouville normal form of equation (30.2.1) is …
14: Bibliography B
  • L. J. Billera, C. Greene, R. Simion, and R. P. Stanley (Eds.) (1996) Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.
  • I. Bloch, M. H. Hull, A. A. Broyles, W. G. Bouricius, B. E. Freeman, and G. Breit (1950) Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.
  • F. Bowman (1958) Introduction to Bessel Functions. Dover Publications Inc., New York.
  • J. P. Boyd and A. Natarov (1998) A Sturm-Liouville eigenproblem of the fourth kind: A critical latitude with equatorial trapping. Stud. Appl. Math. 101 (4), pp. 433–455.
  • W. S. Burnside and A. W. Panton (1960) The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.
  • 15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    A survey is given of the formal spectral theory of second order differential operators, typical results being presented in §1.18(i) through §1.18(viii). The various types of spectra and the corresponding eigenfunction expansions are illustrated by examples. These are based on the Liouville normal form of (1.13.29). A more precise mathematical discussion then follows in §1.18(ix). Eigenvalues and eigenfunctions of T , self-adjoint extensions of with well defined boundary conditions, and utilization of such eigenfunctions for expansion of wide classes of L 2 functions, will be the focus of the remainder of this section. The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. This work is well overviewed by Coddington and Levinson (1955, Ch. 9), and then applied in detail by Titchmarsh (1946), Titchmarsh (1962a), Titchmarsh (1958), and Levitan and Sargsjan (1975) which also connects the Weyl theory to the relevant functional analysis. In parallel, similar, and more general formulations have grown out of functional analysis itself, as in the work of Stone (1990), Rudin (1973), Reed and Simon (1980), Reed and Simon (1975), Reed and Simon (1978), Reed and Simon (1979), Cycon et al. (2008), Dunford and Schwartz (1988, Ch. XIII), Hall (2013, pp. 127-223). Friedman (1990) provides a useful introduction to both approaches; as does the conference proceeding Amrein et al. (2005), overviewing the combination of Sturm–Liouville theory and Hilbert space theory. See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of 51 solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.
    16: Bibliography M
  • L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.
  • S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.
  • S. C. Milne (1997) Balanced Θ 2 3 summation theorems for U ( n ) basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.
  • M. E. Muldoon (1970) Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.
  • M. E. Muldoon (1977) Higher monotonicity properties of certain Sturm-Liouville functions. V. Proc. Roy. Soc. Edinburgh Sect. A 77 (1-2), pp. 23–37.
  • 17: 2.8 Differential Equations with a Parameter
    In Case III f ( z ) has a simple pole at z 0 and ( z z 0 ) 2 g ( z ) is analytic at z 0 . … First we apply the Liouville transformation1.13(iv)) to (2.8.1). … In Case III the approximating equation is … For connection formulas for LiouvilleGreen approximations across these transition points see Olver (1977b, a, 1978). … For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …
    18: 3.8 Nonlinear Equations
    For real functions f ( x ) the sequence of approximations to a real zero ξ will always converge (and converge quadratically) if either: … Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … Initial approximations to the zeros can often be found from asymptotic or other approximations to f ( z ) , or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … … For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the LiouvilleGreen (WKB) approximation, see Segura (2013). …
    19: 15.16 Products
    20: 1.9 Calculus of a Complex Variable
    DeMoivre’s Theorem
    Jordan Curve Theorem
    Cauchy’s Theorem
    Liouville’s Theorem
    Dominated Convergence Theorem