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11: 2.8 Differential Equations with a Parameter
In addition, 𝒱 𝒬 j ( A 1 ) and 𝒱 𝒬 j ( A n ) must be bounded on 𝚫 j ( α j ) . …
12: Bibliography J
  • A. T. James (1964) Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.
  • A. J. E. M. Janssen (2021) Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.
  • 13: Bibliography P
  • R. B. Paris (2002a) Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.
  • T. G. Pedersen (2003) Variational approach to excitons in carbon nanotubes. Phys. Rev. B 67 (7), pp. (073401–1)–(073401–4).
  • E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.
  • 14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    Bounded and Unbounded Linear Operators
    A linear operator T on V is bounded with norm T if … If T is a bounded linear operator on V then its adjoint is the bounded linear operator T such that, for v , w V , … See Brownstein (2000) and Yang and de Llano (1989) for numerical examples, based on variational calculations, and Simon (1976) and Chadan et al. (2003) for rigorous mathematical discussion. … If T is a bounded operator then its spectrum is a closed bounded subset of . …
    15: 2.7 Differential Equations
    For error bounds for (2.7.14) see Olver (1997b, Chapter 7). … For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: … provided that 𝒱 a j , x ( F ) < . …and 𝒱 denotes the variational operator (§2.3(i)). … Assuming also 𝒱 a 1 , a 2 ( F ) < , we have …
    16: Bibliography M
  • J. P. Mills (1926) Table of the ratio: Area to bounding ordinate, for any portion of normal curve. Biometrika 18, pp. 395–400.
  • C. Mortici (2011b) New sharp bounds for gamma and digamma functions. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 57 (1), pp. 57–60.
  • C. Mortici (2013b) Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Modelling 57 (5-6), pp. 1360–1363.
  • R. J. Muirhead (1978) Latent roots and matrix variates: A review of some asymptotic results. Ann. Statist. 6 (1), pp. 5–33.
  • M. E. Muldoon (1981) The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.
  • 17: Bibliography S
  • F. W. Schäfke and A. Finsterer (1990) On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.
  • C. Schwartz (1961) Variational calculations of scattering. Ann. Phys. 16, pp. 36–50.
  • J. Segura (2001) Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.
  • J. Segura (2011) Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374 (2), pp. 516–528.
  • B. Simon (1976) The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.