solutions analytic at two singularities

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11: 1.10 Functions of a Complex Variable

… ►Lastly, if $a_n\ne 0$ 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 $ℂ$ with at most one exception. … ►has a unique solution $z=F(w)$ analytic at $w=w_0$, and … ►Furthermore, if $g(z)$ is analytic at $z_0$, then … ►Also, if in addition $g(z)$ is analytic at $z_0$, then …

12: 10.2 Definitions

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§10.2(i) Bessel’s Equation

… ►This differential equation has a regular singularity at $z=0$ with indices $\pm \nu$, and an irregular singularity at $z=\mathrm{\infty }$ of rank $1$; compare §§2.7(i) and 2.7(ii). … ►This solution of (10.2.1) is an analytic function of $z\in ℂ$, except for a branch point at $z=0$ when $\nu$ is not an integer. … ►Each solution has a branch point at $z=0$ for all $\nu \in ℂ$. … ►

§10.2(iii) Numerically Satisfactory Pairs of Solutions

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13: Bibliography B

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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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The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, pp. 183–214.

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External Links:

ISSN 0075-4102, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

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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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External Links:

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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An analytic continuation of the hypergeometric series. SIAM J. Math. Anal. 18 (3), pp. 884–889.

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External Links:

ISSN 0036-1410, Document, MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§15.15, §15.19(i).

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14: 31.15 Stieltjes Polynomials

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§31.15(i) Definitions

►Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). …There exist at most $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $V(z)$ of degree not exceeding $N-2$ such that for $\mathrm{\Phi }(z)=V(z)$, (31.15.1) has a polynomial solution $w=S(z)$ of degree $n$. … ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $𝐦=(m_1,m_2,\mathrm{\dots },m_{N-1})$, 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 $j=1,2,\mathrm{\dots },N-1$. … ►For further details and for the expansions of analytic functions in this basis see Volkmer (1999).

15: 15.2 Definitions and Analytical Properties

§15.2 Definitions and Analytical Properties

… ►again with analytic continuation for other values of $z$, and with the principal branch defined in a similar way. … ►

§15.2(ii) Analytic Properties

… ►As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. … ►(Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does $𝐅(a,b;c;z)$, which is analytic at $c=0,-1,-2,\mathrm{\dots }$.) …

16: 18.40 Methods of Computation

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Stieltjes Inversion via (approximate) Analytic Continuation

… ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … ►This is a challenging case as the desired $w^{\mathrm{RCP}}(x)$ on $[-1,1]$ has an essential singularity at $x=-1$. … ►Achieving precisions at this level shown above requires higher than normal computational precision, see Gautschi (2009). …