complex%20variables
(0.003 seconds)
21—30 of 34 matching pages
21: Bibliography G
…
►
Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
…
►
Efficient computation of the complex error function.
SIAM J. Numer. Anal. 7 (1), pp. 187–198.
…
►
Variable-precision recurrence coefficients for nonstandard orthogonal polynomials.
Numer. Algorithms 52 (3), pp. 409–418.
…
►
Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
…
►
Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
…
22: 36.5 Stokes Sets
…
►The Stokes set consists of the rays in the complex
-plane.
…
►This part of the Stokes set connects two complex saddles.
…
►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets.
In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles.
The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions.
…
23: 10.3 Graphics
…
►
§10.3(i) Real Order and Variable
… ►§10.3(ii) Real Order, Complex Variable
… ► … ► ►§10.3(iii) Imaginary Order, Real Variable
…24: Bibliography L
…
►
The solutions of the Mathieu equation with a complex variable and at least one parameter large.
Trans. Amer. Math. Soc. 36 (3), pp. 637–695.
…
►
Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
…
►
An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
…
►
Complex Variables.
Holden-Day Inc., San Francisco, CA.
…
►
Discrete-variable representations and their utilization.
In Advances in Chemical Physics,
pp. 263–310.
…
25: 11.6 Asymptotic Expansions
26: Bibliography
…
►
Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
…
►
On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
…
►
Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable.
2nd edition, McGraw-Hill Book Co., New York.
…
►
Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
…
►
Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
…
27: 12.10 Uniform Asymptotic Expansions for Large Parameter
…
►In this section we give asymptotic expansions of PCFs for large values of the parameter that are uniform with respect to the variable
, when both and
are real.
These expansions follow from Olver (1959), where detailed information is also given for complex variables.
…
►Here bars do not denote complex conjugates; instead
…
►(For complex values of and see Olver (1959).)
…
►The variable
is defined by
…
28: 19.36 Methods of Computation
…
►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18).
…
►Because may be real and negative, or even complex, care is needed to ensure , and similarly for and .
…
►Complex values of the variables are allowed, with some restrictions in the case of that are sufficient but not always necessary.
…
►For computation of and with complex
see Fettis and Caslin (1969) and Morita (1978).
…
►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).
…
29: 9.9 Zeros
…
►However, and each have an infinite number of complex zeros.
They lie in the sectors and , and are denoted by , , respectively, in the former sector, and by , , in the conjugate sector, again arranged in ascending order of absolute value (modulus) for See §9.3(ii) for visualizations.
►For the distribution in of the zeros of , where is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014).
…
►If is regarded as a continuous variable, then
…
►Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of and in the upper half plane.
…
30: Bibliography D
…
►
Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
►
Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
.
Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
…
►
Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
…
►
Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
…
►
Coulomb wave functions with complex values of the variable and the parameters.
J. Math. Phys. 40 (12), pp. 6145–6166.