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61: Bibliography K

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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).

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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

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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. J. Comput. Appl. Math. 205 (1), pp. 186–206.

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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:

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62: Bibliography L

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Heun’s equation with nearby singularities. Proc. Roy. Soc. London Ser. A 455, pp. 4347–4361.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§31.13, §31.17(ii).

Encodings:

BibTeX

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Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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63: 1.8 Fourier Series

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. …

64: 3.7 Ordinary Differential Equations

… ►For classification of singularities of (3.7.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …

65: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta =0$, and the Maclaurin series expansion of $c_k(\eta )$ is given by … ►A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by …

66: 18.39 Applications in the Physical Sciences

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. For $Z=+1$ the weight function, red curve, has an essential singularity at $x=-1$, as all derivatives vanish as $x\to -1+$; the green curve is ${\int }_{-1}^x{w}^{\mathrm{CP}}(y)dy$, to be compared with its histogram approximation in §18.40(ii). For $Z=-1$ the weight function, blue curve, is non-zero at $x=-1$, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as $k\to \mathrm{\infty }$, $x_k\to -1-$. Magnify

… ►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to -1-$. …

67: 1.4 Calculus of One Variable

… ►A removable singularity of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. … …

68: 2.3 Integrals of a Real Variable

… ►Other types of singular behavior in the integrand can be treated in an analogous manner. … ►For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). … ►it is free from singularity at $t=\alpha$. …

69: 1.16 Distributions

… ►(If a distribution is not regular, it is called singular.) More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: … ►The Dirac delta distribution is singular. …

70: 2.9 Difference Equations

… ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). …