isolated essential

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11—20 of 21 matching pages

11: 18.40 Methods of Computation

… ►This is a challenging case as the desired

w^{\mathrm{RCP}}(x)

[-1,1]

has an essential singularity at

x=-1

. …

12: 18.37 Classical OP’s in Two or More Variables

… ►In one variable they are essentially ultraspherical, Jacobi, continuous

q

-ultraspherical, or Askey–Wilson polynomials. …

14: 1.4 Calculus of One Variable

… ►Ismail (2005, p 5) refers to these

x_{n}

as isolated atoms. …

A. A. Kapaev (1991) Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187, pp. 139–170 (Russian).

ⓘ

Notes:: English translation: J. Math. Sci. 73(1995), no. 4, pp. 500–517
External Links:: ISSN 0373-2703, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt
Referenced by:: §32.12(ii).
Encodings:: BibTeX

…

16: 32.10 Special Function Solutions

… ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since

\mbox{P}_{\mbox{\scriptsize VI}}

with

\alpha=\beta=\gamma=0

and

\delta=\tfrac{1}{2}

does not admit an algebraic first integral of the form

P(z,w,w^{\prime},C)=0

, with

C

a constant. …

17: 2.4 Contour Integrals

… ►However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. …

18: 19.36 Methods of Computation

… ►All cases of

R_{F}

R_{C}

R_{J}

, and

R_{D}

are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). …

19: 9.13 Generalized Airy Functions

… ►The function on the right-hand side is recessive in the sector

-(2j-1)\pi/m\leq\operatorname{ph}z\leq(2j+1)\pi/m

, and is therefore an essential member of any numerically satisfactory pair of solutions in this region. …

20: 18.35 Pollaczek Polynomials

… ►indicating the presence of essential singularities. …