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41: 2.10 Sums and Sequences

… ►For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a). … ►However, if $r$ is finite and $f(z)$ has algebraic or logarithmic singularities on $|z|=r$, then Darboux’s method is usually easier to apply. … ►in the neighborhood of each singularity $z_j$, again with ${\sigma }_j>0$. … ►The singularities of $f(z)$ on the unit circle are branch points at $z={\mathrm{e}}^{\pm \mathrm{i}\alpha }$. … ►For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005). …

42: Bibliography B

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Unfolding the high orders of asymptotic expansions with coalescing saddles: Singularity theory, crossover and duality. Proc. Roy. Soc. London Ser. A 443, pp. 107–126.

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External Links:

ISSN 0962-8444, FullText, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§36.12(i), §36.12(ii), §36.12(iii).

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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.

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Referenced by:

§36.14(i).

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Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, pp. 353–370.

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External Links:

Document, MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§2.3(v).

Encodings:

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Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities. J. Math. Mech. 17, pp. 533–559.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2), pp. 422–450.

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ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.15(iv), §14.15(iv), §14.26, §2.8(vi).

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43: 1.13 Differential Equations

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§1.13(vi) Singularities

►For classification of singularities of (1.13.1) and expansions of solutions in the neighborhoods of singularities, see §2.7. …

44: 2.4 Contour Integrals

… ►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). …

45: 10.47 Definitions and Basic Properties

… ►Equations (10.47.1) and (10.47.2) each have a regular singularity at $z=0$ with indices $n$, $-n-1$, and an irregular singularity at $z=\mathrm{\infty }$ of rank $1$; compare §§2.7(i)–2.7(ii). …

46: 20.13 Physical Applications

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

47: 28.7 Analytic Continuation of Eigenvalues

… ►The only singularities are algebraic branch points, with $a_n\left(q\right)$ and $b_n\left(q\right)$ finite at these points. …

48: Bibliography F

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

Referenced by:

§2.10(iv).

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49: Bibliography W

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Asymptotic expansions of Fourier transforms of functions with logarithmic singularities. J. Math. Anal. Appl. 64 (1), pp. 173–180.

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External Links:

Document, MathReview (B. L. Gupta), ZentralBlatt

Referenced by:

§2.3(iv).

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Asymptotic expansions of Hankel transforms of functions with logarithmic singularities. Comput. Math. Appl. 3 (4), pp. 271–286.

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External Links:

ISSN 0886-9561, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(v).

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50: 15.2 Definitions and Analytical Properties

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§15.2(ii) Analytic Properties

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