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Liouville–Green approximation theorem

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11: Bibliography L
  • J. Lagrange (1770) Démonstration d’un Théoréme d’Arithmétique. Nouveau Mém. Acad. Roy. Sci. Berlin, pp. 123–133 (French).
  • L. Lorch, M. E. Muldoon, and P. Szegő (1970) Higher monotonicity properties of certain Sturm-Liouville functions. III. Canad. J. Math. 22, pp. 1238–1265.
  • L. Lorch, M. E. Muldoon, and P. Szegő (1972) Higher monotonicity properties of certain Sturm-Liouville functions. IV. Canad. J. Math. 24, pp. 349–368.
  • L. Lorch and P. Szegő (1963) Higher monotonicity properties of certain Sturm-Liouville functions.. Acta Math. 109, pp. 55–73.
  • Y. L. Luke (1968) Approximations for elliptic integrals. Math. Comp. 22 (103), pp. 627–634.
  • 12: 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). …
    13: Bibliography M
  • J. W. Miles (1975) Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.
  • S. C. Milne (1988) A q -analog of the Gauss summation theorem for hypergeometric series in U ( n ) . Adv. in Math. 72 (1), pp. 59–131.
  • 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.
  • 14: Bibliography G
  • M. B. Green, J. H. Schwarz, and E. Witten (1988a) Superstring Theory: Introduction, Vol. 1. 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
  • M. B. Green, J. H. Schwarz, and E. Witten (1988b) Superstring Theory: Loop Amplitudes, Anomalies and Phenomenolgy, Vol. 2. 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
  • C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.
  • D. H. Greene and D. E. Knuth (1982) Mathematics for the Analysis of Algorithms. Progress in Computer Science, Vol. 1, Birkhäuser Boston, Boston, MA.
  • A. J. Guttmann and T. Prellberg (1993) Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.
  • 15: 27.6 Divisor Sums
    27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
    16: 27.7 Lambert Series as Generating Functions
    27.7.6 n = 1 λ ( n ) x n 1 - x n = n = 1 x n 2 .
    17: 32.3 Graphics
    See accompanying text
    Figure 32.3.3: w k ( x ) for - 12 x 0.73 and k = 1.85185 3 , 1.85185 5 . The two graphs are indistinguishable when x exceeds - 5.2 , approximately. … Magnify
    See accompanying text
    Figure 32.3.7: u k ( x ; - 1 2 ) for - 12 x 4 with k = 0.33554 691 , 0.33554 692 . …The parabolas u 2 + 1 2 x = 0 , u 2 + 1 6 x = 0 are shown in black and green, respectively. Magnify
    See accompanying text
    Figure 32.3.8: u k ( x ; 1 2 ) for - 12 x 4 with k = 0.47442 , 0.47443 . …The curves u 2 + 1 3 x ± 1 6 x 2 + 12 = 0 are shown in green and black, respectively. Magnify
    See accompanying text
    Figure 32.3.9: u k ( x ; 3 2 ) for - 12 x 4 with k = 0.38736 , 0.38737 . …The curves u 2 + 1 3 x ± 1 6 x 2 + 24 = 0 are shown in green and black, respectively. Magnify
    See accompanying text
    Figure 32.3.10: u k ( x ; 5 2 ) for - 12 x 4 with k = 0.24499 2 , 0.24499 3 . …The curves u 2 + 1 3 x ± 1 6 x 2 + 36 = 0 are shown in green and black, respectively. Magnify
    18: Bibliography P
  • R. Piessens (1984a) Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.
  • R. Piessens and S. Ahmed (1986) Approximation for the turning points of Bessel functions. J. Comput. Phys. 64 (1), pp. 253–257.
  • M. J. D. Powell (1967) On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.
  • P. J. Prince (1975) Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.
  • J. D. Pryce (1993) Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.
  • 19: 1.9 Calculus of a Complex Variable
    DeMoivre’s Theorem
    Jordan Curve Theorem
    Cauchy’s Theorem
    Liouville’s Theorem
    Dominated Convergence Theorem
    20: 30.2 Differential Equations
    The Liouville normal form of equation (30.2.1) is …