1—10 of 66 matching pages
1: 9.15 Mathematical Applications
… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
2: 10.72 Mathematical Applications
Simple Turning Points►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)). … ►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as . … ►In (10.72.1) assume and depend continuously on a real parameter , has a simple zero and a double pole , except for a critical value , where . …
3: 13.27 Mathematical Applications
… ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …
4: 27.18 Methods of Computation: Primes
… ►Two simple algorithms for proving primality require a knowledge of all or part of the factorization of , or both; see Crandall and Pomerance (2005, §§4.1–4.2). …
5: 5.2 Definitions
… ►It is a meromorphic function with no zeros, and with simple poles of residue at . is entire, with simple zeros at . … is meromorphic with simple poles of residue at . …
6: 31.6 Path-Multiplicative Solutions
… ►This denotes a set of solutions of (31.2.1) with the property that if we pass around a simple closed contour in the -plane that encircles and once in the positive sense, but not the remaining finite singularity, then the solution is multiplied by a constant factor . …
7: Bibliography D
A simple sum formula for Clebsch-Gordan coefficients.
Lett. Math. Phys. 5 (3), pp. 207–211.
Novel identities for simple
J. Mathematical Phys. 16, pp. 318–319.
Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane.
SIAM J. Math. Anal. 25 (2), pp. 322–353.
Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions.
Stud. Appl. Math. 107 (3), pp. 293–323.
Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point.
Anal. Appl. (Singap.) 12 (4), pp. 385–402.
8: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations… ►If and , then is a simple zero of . … ►If is a simple zero, then the iteration converges locally and quadratically. … ►If the wanted zero is simple, then the method converges locally with order of convergence . … ►Then the sensitivity of a simple zero to changes in is given by …
9: 8.27 Approximations
DiDonato (1978) gives a simple approximation for the function (which is related to the incomplete gamma function by a change of variables) for real and large positive . This takes the form , approximately, where and is shown to produce an absolute error as .