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1: 36.5 Stokes Sets
For z = 0 , the set consists of the two curves …where x ± are the two smallest positive roots of the equation … For z > 0 the Stokes set has two sheets. … This part of the Stokes set connects two complex saddles. … In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …
2: Bibliography C
  • K. Chadan, N. N. Khuri, A. Martin, and T. T. Wu (2003) Bound states in one and two spatial dimensions. J. Math. Phys. 44 (2), pp. 406–422.
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 3: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near T c . Phys. Rev. Lett. 31, pp. 1409–1411.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • P. Boalch (2006) The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, pp. 183–214.
  • 4: 18.40 Methods of Computation
    Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. … There are many ways to implement these first two steps, noting that the expressions for α n and β n of equation (18.2.30) are of little practical numerical value, see Gautschi (2004) and Golub and Meurant (2010). … Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . … The quadrature points and weights can be put to a more direct and efficient use. … In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …
    5: Bibliography L
  • V. Laĭ (1994) The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).
  • W. Lay and S. Yu. Slavyanov (1998) The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • N. Levinson (1974) More than one third of zeros of Riemann’s zeta-function are on σ = 1 2 . Advances in Math. 13 (4), pp. 383–436.
  • E. R. Love (1972b) Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.
  • 6: Bibliography R
  • J. T. Ratnanather, J. H. Kim, S. Zhang, A. M. J. Davis, and S. K. Lucas (2014) Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
  • 7: Bibliography F
  • FDLIBM (free C library)
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • C. L. Frenzen (1987a) Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18 (3), pp. 890–896.
  • C. L. Frenzen (1992) Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.
  • G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
  • 8: 9.7 Asymptotic Expansions
    Numerical values of χ ( n ) are given in Table 9.7.1 for n = 1 ( 1 ) 20 to 2D. …
    §9.7(iii) Error Bounds for Real Variables
    In (9.7.7) and (9.7.8) the n th error term is bounded in magnitude by the first neglected term multiplied by χ ( n + σ ) + 1 where σ = 1 6 for (9.7.7) and σ = 0 for (9.7.8), provided that n 0 in the first case and n 1 in the second case. …
    §9.7(iv) Error Bounds for Complex Variables
    provided that n 0 , σ = 1 6 for (9.7.5) and n 1 , σ = 0 for (9.7.6). …
    9: 8.17 Incomplete Beta Functions
    Throughout §§8.17 and 8.18 we assume that a > 0 , b > 0 , and 0 x 1 . …
    8.17.4 I x ( a , b ) = 1 I 1 x ( b , a ) .
    With a > 0 , b > 0 , and 0 < x < 1 , … For x > ( a + 1 ) / ( a + b + 2 ) or 1 x < ( b + 1 ) / ( a + b + 2 ) , more rapid convergence is obtained by computing I 1 x ( b , a ) and using (8.17.4). …
    8.17.24 I x ( m , n ) = ( 1 x ) n j = m ( n + j 1 j ) x j , m , n positive integers; 0 x < 1 .
    10: Bibliography I
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • M. E. H. Ismail, J. Letessier, G. Valent, and J. Wimp (1990) Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.
  • M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.
  • M. E. H. Ismail (2000b) More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.
  • M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.