limit points (or limiting points)

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21: 15.2 Definitions and Analytical Properties

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§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 $\mathrm{\infty }$. … ►Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … ►

15.2.5 $$F(\genfrac{}{}{0pt}{}{-m,b}{-m-\mathrm{\ell }};z)=\underset{c\to -m-\mathrm{\ell }}{lim}\left(\underset{a\to -m}{lim}F(\genfrac{}{}{0pt}{}{a,b}{c};z)\right),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\mathrm{\ell }$: integer and $m$: integer

Referenced by:

§15.10(i), §15.2(ii), §15.2(ii), §15.4(i), §15.4(iii)

Permalink:

http://dlmf.nist.gov/15.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

… ►For comparison of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. …

22: 11.6 Asymptotic Expansions

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23: 19.2 Definitions

… ►The paths of integration are the line segments connecting the limits of integration. The integral for $E(\varphi ,k)$ is well defined if $k^2={\sin }^2\varphi =1$, and the Cauchy principal value (§1.4(v)) of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is taken if $1-{\alpha }^2{\sin }^2\varphi$ vanishes at an interior point of the integration path. …The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points ${\alpha }^2=0,k^2,1$. … ►with a branch point at $k=0$ and principal branch $|\mathrm{ph}k|\le \pi$. …

24: Bibliography D

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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.

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External Links:

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

Referenced by:

§14.26, §2.8(vi).

Encodings:

BibTeX

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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.

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External Links:

ISSN 0036-1410, Document, MathReview (H. C. Howard), ZentralBlatt

Referenced by:

§2.8(vi).

Encodings:

BibTeX

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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§2.8(vi), §8.12, §8.12.

Encodings:

BibTeX

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

Encodings:

BibTeX

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

Encodings:

BibTeX

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25: 2.8 Differential Equations with a Parameter

… ►Zeros of $f(z)$ are also called turning points. … ►

§2.8(ii) Case I: No Transition Points

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§2.8(iii) Case II: Simple Turning Point

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§2.8(v) Multiple and Fractional Turning Points

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§2.8(vi) Coalescing Transition Points

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26: 2.3 Integrals of a Real Variable

… ►However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of $p^{\prime }(t)$ because $p(t)$ changes relatively slowly at these stationary points. … ►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. … ►

If the limit $p(b)$ of $p(t)$ as $t\to b-$ is finite, then each of the functions

2.3.21 $$P_s(t)={\left(\frac{1}{p^{\prime }(t)}\frac{d}{dt}\right)}^s\frac{q(t)}{p^{\prime }(t)},$$ $s=0,1,2,\mathrm{\dots }$,

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Defines:

$P_s(t)$: function (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p(t)$: real function and $q(t)$: function

Permalink:

http://dlmf.nist.gov/2.3.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(iv), §2.3 and Ch.2

tends to a finite limit $P_s(b)$.

… ►where $a$ and $b$ are functions of $\alpha$ chosen in such a way that $t=0$ corresponds to $w=0$, and the stationary points $t=\alpha$ and $w=a$ correspond. … ►We replace the limit $\kappa$ by $\mathrm{\infty }$ and integrate term-by-term: …

27: 7.21 Physical Applications

… ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. …

28: 2.4 Contour Integrals

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§2.4(iv) Saddle Points

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§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

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§2.4(vi) Other Coalescing Critical Points

… ►For a coalescing saddle point, a pole, and a branch point see Ciarkowski (1989). For many coalescing saddle points see §36.12. …

29: 10.2 Definitions

… ►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. … ►When $\nu$ is an integer the right-hand side is replaced by its limiting value: …Whether or not $\nu$ is an integer $Y_{\nu }\left(z\right)$ has a branch point at $z=0$. … ►

Bessel Functions of the Third Kind (Hankel Functions)

… ►Each solution has a branch point at $z=0$ for all $\nu \in ℂ$. …

30: 13.14 Definitions and Basic Properties

… ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►

13.14.7 $$\underset{2\mu \to -n-1}{lim}\frac{M_{\kappa ,\mu }\left(z\right)}{\mathrm{\Gamma }\left(2\mu +1\right)}=\frac{{\left(-\frac{1}{2}n-\kappa \right)}_{n+1}}{(n+1)!}{M}_{\kappa ,\frac{1}{2}(n+1)}\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{-\frac{1}{2}n}\sum _{s=n+1}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{2}n-\kappa \right)}_s}{\mathrm{\Gamma }\left(s-n\right)s!}{z}^s.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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§13.14(iii) Limiting Forms as $z\to 0$

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§13.14(iv) Limiting Forms as $z\to \mathrm{\infty }$

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