behavior at singularities
1—10 of 11 matching pages
§14.21(iii) Properties… ►This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
§14.20(iii) Behavior as…
… ►With denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of and near the singularity at is given by the following convergent series: …
… ►For extensions of the Euler–Maclaurin formula to functions with singularities at or (or both) see Sidi (2004, 2012b, 2012a). … ►We seek the behavior as . … ►
►The singularities of on the unit circle are branch points at
To match the limiting behavior of
at these points we set
On the circle , the function has a finite number of singularities, and at each singularity , say,
where is a positive constant.
… ►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at , and the Maclaurin series expansion of is given by … ►For the asymptotic behavior of as see Dunster et al. (1998) and Olde Daalhuis (1998c). … ►A different type of uniform expansion with coefficients that do not possess a removable singularity at is given by …
… ►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … ►If both and are analytic at , then is a regular singularity (or singularity of the first kind). … ►The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … ►The transformed differential equation either has a regular singularity at , or its characteristic equation has unequal roots. … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …
Lebesgue Constants… ►(1.8.10) continues to apply if either or or both are infinite and/or has finitely many singularities in , provided that the integral converges uniformly (§1.5(iv)) at , and the singularities for all sufficiently large . … ►Let be an absolutely integrable function of period , and continuous except at a finite number of points in any bounded interval. …at every point at which has both a left-hand derivative (that is, (1.4.4) applies when ) and a right-hand derivative (that is, (1.4.4) applies when ). The convergence is non-uniform, however, at points where ; see §6.16(i). …
Singularities in Waves and Rays.
In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.),
Vol. 35, pp. 453–543.
Uniform asymptotic expansions of integrals with stationary point near algebraic singularity.
Comm. Pure Appl. Math. 19, pp. 353–370.
Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities.
J. Math. Mech. 17, pp. 533–559.
Asymptotic behavior of the Pollaczek polynomials and their zeros.
Stud. Appl. Math. 96, pp. 307–338.
The behavior at unit argument of the hypergeometric function
SIAM J. Math. Anal. 18 (5), pp. 1227–1234.
… ►Other types of singular behavior in the integrand can be treated in an analogous manner. … ►Without loss of generality, we assume that this minimum is at the left endpoint . … ►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. … ►For extensions to oscillatory integrals with more general -powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). … ►it is free from singularity at . …