isolated singularity

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11: 36.14 Other Physical Applications

… ►These are the structurally stable focal singularities (envelopes) of families of rays, on which the intensities of the geometrical (ray) theory diverge. Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. …

12: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …

13: 31.18 Methods of Computation

… ►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …

14: 16.8 Differential Equations

… ►

§16.8(i) Classification of Singularities

… ►All other singularities are irregular. … … ►In each case there are no other singularities. … ►

§16.8(iii) Confluence of Singularities

…

15: 30.2 Differential Equations

… ►This equation has regular singularities at

z=\pm 1

with exponents

\pm\frac{1}{2}\mu

and an irregular singularity of rank 1 at

z=\infty

(if

\gamma\neq 0

). … …

16: Mark J. Ablowitz

… ►ODEs which do not have moveable branch point singularities. …

17: 31.15 Stieltjes Polynomials

… ►

31.15.2

\sum_{j=1}^{N}\frac{\gamma_{j}/2}{z_{k}-a_{j}}+\sum_{\begin{subarray}{c}j=1\\ j\neq k\end{subarray}}^{n}\frac{1}{z_{k}-z_{j}}=0,

k=1,2,\dots,n

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(ii)
Permalink:: http://dlmf.nist.gov/31.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.3

\sum_{j=1}^{N}\frac{\gamma_{j}}{t_{k}-a_{j}}+\sum_{j=1}^{n-1}\frac{1}{t_{k}-z_% {j}^{\prime}}=0.

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $t_{k}$ : zero of $V(z)$
Permalink:: http://dlmf.nist.gov/31.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.6

a_{j}<a_{j+1},

j=1,2,\dots,N-1

ⓘ

Symbols:: $j$ : nonnegative integer, $a$ : complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.7

q_{j}=\gamma_{j}\sum_{k=1}^{n}\frac{1}{z_{k}-a_{j}},

j=1,2,\dots,N

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Permalink:: http://dlmf.nist.gov/31.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(ii), §31.15 and Ch.31

… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index

\mathbf{m}=(m_{1},m_{2},\dots,m_{N-1})

, where each

m_{j}

is a nonnegative integer, there is a unique Stieltjes polynomial with

m_{j}

zeros in the open interval

(a_{j},a_{j+1})

for each

j=1,2,\dots,N-1

. …

18: 2.7 Differential Equations

… ►

§2.7(i) Regular Singularities: Fuchs–Frobenius Theory

… ►Other points

z_{0}

are singularities of the differential equation. …All other singularities are classified as irregular. … ►

§2.7(ii) Irregular Singularities of Rank 1

… ►Thus a regular singularity has rank 0. …

19: 15.11 Riemann’s Differential Equation

… ►

§15.11(i) Equations with Three Singularities

►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … ►Cases in which there are fewer than three singularities are included automatically by allowing the choice

\{0,1\}

for exponent pairs. … ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …

20: 10.72 Mathematical Applications

… ►These expansions are uniform with respect to

z

, including the turning point

z_{0}

and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … ►These asymptotic expansions are uniform with respect to

z

, including cut neighborhoods of

z_{0}

, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. …