isolated essential singularity

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1: 1.10 Functions of a Complex Variable

… ►Then

z=z_{0}

is an isolated singularity of

f(z)

. …Lastly, if

a_{n}\not=0

for infinitely many negative

n

, then

z_{0}

is an isolated essential singularity. … ► … ► … ►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in

\mathbb{C}

with at most one exception. …

2: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which

F

is a rational function of

w

and

\ifrac{\mathrm{d}w}{\mathrm{d}z}

, and is locally analytic in

z

, that is, analytic except for isolated singularities in

\mathbb{C}

. In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions. An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however. …

3: 31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

4: 14.32 Methods of Computation

… ►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. …

5: 2.4 Contour Integrals

… ►However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. … ►The problems sketched in §§2.3(v) and 2.4(v) involve only two of many possibilities for the coalescence of endpoints, saddle points, and singularities in integrals associated with the special functions. …For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. … ►For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998). …

6: Bibliography K

… ►

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

… ►

D. Karp and S. M. Sitnik (2007) Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. J. Comput. Appl. Math. 205 (1), pp. 186–206.

ⓘ

External Links:: ISSN 0377-0427, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §19.12, §19.12, §19.36(iv), §19.36(iv).
Encodings:: BibTeX

…

7: 34.12 Physical Applications

… ►The angular momentum coupling coefficients (

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols) are essential in the fields of nuclear, atomic, and molecular physics. …

8: 31.12 Confluent Forms of Heun’s Equation

… ►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … ►This has regular singularities at

z=0

and

1

, and an irregular singularity of rank 1 at

z=\infty

. … ►This has irregular singularities at

z=0

and

\infty

, each of rank

1

. … ►This has a regular singularity at

z=0

, and an irregular singularity at

\infty

of rank

2

. … ►This has one singularity, an irregular singularity of rank

3

z=\infty

. …

9: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with

N+1

regular singularities at

z=a_{j}

j=1,2,\dots,N

, and at

\infty

, is given by ►

31.14.1

{\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},

\sum_{j=1}^{N}q_{j}=0

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(i)
Permalink:: http://dlmf.nist.gov/31.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14 and Ch.31

►The exponents at the finite singularities

a_{j}

are

\{0,{1-\gamma_{j}}\}

and those at

\infty

are

\{\alpha,\beta\}

, where …The three sets of parameters comprise the singularity parameters

a_{j}

, the exponent parameters

\alpha,\beta,\gamma_{j}

, and the

N-2

free accessory parameters

q_{j}

. … ►

31.14.3

w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $W(z)$ : solution
Permalink:: http://dlmf.nist.gov/31.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

…

10: 31.1 Special Notation

… ►Sometimes the parameters are suppressed.