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21: 14.2 Differential Equations

… ►Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\mathrm{\infty }$, with exponent pairs $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, and $\{\nu +1,-\nu \}$, respectively; compare §2.7(i). … ►When $\mathrm{\Re }\mu \ge 0$ and $\mathrm{\Re }\nu \ge -\frac{1}{2}$, $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\mathrm{\infty }$, respectively. …

22: 16.21 Differential Equation

… ►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\mathrm{\infty }$. When $p=q$ the only singularities of (16.21.1) are regular singularities at $z=0$, ${(-1)}^{p-m-n}$, and $\mathrm{\infty }$. …

23: 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 $z=\mathrm{\infty }$ (if $\gamma \ne 0$). …

24: 13.14 Definitions and Basic Properties

… ►It has a regular singularity at the origin with indices $\frac{1}{2}\pm \mu$, and an irregular singularity at infinity of rank one. … ►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 $z=\mathrm{\infty }$. …

25: 28.2 Definitions and Basic Properties

… ►This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\mathrm{\infty }$. …

26: 3.11 Approximation Techniques

… ►A general procedure is to approximate $F$ by a rational function $R$ (vanishing at infinity) and then approximate $f$ by $r={ℒ}^{-1}R$. …

27: 26.10 Integer Partitions: Other Restrictions

… ►

26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

…

28: 3.5 Quadrature

… ►We choose $s=1$ so that $f(\zeta )=O\left(1\right)$ at infinity. …

29: 5.9 Integral Representations

… ►where the contour begins at $-\mathrm{\infty }$, circles the origin once in the positive direction, and returns to $-\mathrm{\infty }$. …

30: 25.5 Integral Representations

… ►where the integration contour is a loop around the negative real axis; it starts at $-\mathrm{\infty }$, encircles the origin once in the positive direction without enclosing any of the points $z=\pm 2\pi \mathrm{i}$, $\pm 4\pi \mathrm{i}$, …, and returns to $-\mathrm{\infty }$. …