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1: Bibliography S
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • 2: 25.12 Polylogarithms
    §25.12(i) Dilogarithms
    §25.12(ii) Polylogarithms
    For each fixed complex s the series defines an analytic function of z for | z | < 1 . …For other values of z , Li s ( z ) is defined by analytic continuation. … and …
    3: 3.8 Nonlinear Equations
    §3.8 Nonlinear Equations
    This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: …
    §3.8(v) Zeros of Analytic Functions
    Newton’s rule is the most frequently used iterative process for accurate computation of real or complex zeros of analytic functions f ( z ) . …
    §3.8(vi) Conditioning of Zeros
    4: William P. Reinhardt
    Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics. … Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. …
  • In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.
    5: Peter L. Walker
    Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …
  • 6: 18.40 Methods of Computation
    Stieltjes Inversion via (approximate) Analytic Continuation
    The question is then: how is this possible given only F N ( z ) , rather than F ( z ) itself? F N ( z ) often converges to smooth results for z off the real axis for z at a distance greater than the pole spacing of the x n , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to F N ( z ) and evaluating these on the real axis in regions of higher pole density that those of the approximating function. Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . … H ( x ) being the Heaviside step-function, see (1.16.13). … The example chosen is inversion from the α n , β n for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). …
    7: Bibliography M
  • A. I. Markushevich (1983) The Theory of Analytic Functions: A Brief Course. “Mir”, Moscow.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • R. Mehrem, J. T. Londergan, and M. H. Macfarlane (1991) Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.
  • D. S. Mitrinović (1970) Analytic Inequalities. Springer-Verlag, New York.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • 8: Bibliography B
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley and J. Nayler (1935) A short table of the functions Ki n ( x ) , from n = 1 to n = 16 . Phil. Mag. Series 7 20, pp. 343–347.
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • W. Bühring (1988) An analytic continuation formula for the generalized hypergeometric function. SIAM J. Math. Anal. 19 (5), pp. 1249–1251.
  • 9: 19.36 Methods of Computation
    When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …where the elementary symmetric functions E s are defined by (19.19.4). … Alternatively, the first duplication is done analytically as in Carlson and FitzSimons (2000), where further information can be found. …
    §19.36(iii) Via Theta Functions
    For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …
    10: 3.4 Differentiation
    §3.4(ii) Analytic Functions
    If f can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …where C is a simple closed contour described in the positive rotational sense such that C and its interior lie in the domain of analyticity of f , and x 0 is interior to C . … As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands. …