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21: Errata

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Pochhammer and $q$-Pochhammer symbols in several equations now correctly link to their definitions.

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Linkage of mathematical symbols to their definitions were corrected or improved.

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In many cases, the links from mathematical symbols to their definitions were corrected or improved. These links were also enhanced with ‘tooltip’ feedback, where supported by the user’s browser.

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It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$ is valid for $h$ $(\in ℂ)$; originally it was unnecessarily restricted to $[0,1]$.

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The definition of the notation $F(z_0{\mathrm{e}}^{2k\pi \mathrm{i}})$ was added in Common Notations and Definitions.

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22: 1.14 Integral Transforms

… ►Suppose $f(t)$ and $g(t)$ are absolutely and square integrable on $[0,\mathrm{\infty })$, then … ►

§1.14(iii) Laplace Transform

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§1.14(iv) Mellin Transform

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§1.14(v) Hilbert Transform

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§1.14(vi) Stieltjes Transform

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23: 26.9 Integer Partitions: Restricted Number and Part Size

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§26.9(i) Definitions

… ►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer $k$. … ►

26.9.4 $${\left[\genfrac{}{}{0.0pt}{}{m}{n}\right]}_q=\prod _{j=1}^n\frac{1-q^{m-n+j}}{1-q^j},$$ $n\ge 0$,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.9(ii), §26.9 and Ch.26

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24: 21.7 Riemann Surfaces

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21.7.2 $$\tilde{P}(\tilde{\lambda },\tilde{\mu },\tilde{\eta })=0,$$

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Defines:

$\tilde{P}(\tilde{\lambda },\tilde{\mu })$: polynomial (locally)

Permalink:

http://dlmf.nist.gov/21.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(i), §21.7 and Ch.21

… ►On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$, such that their intersection indices satisfy …

25: 23.20 Mathematical Applications

… ►There is a unique point $z_0\in [{\omega }_1,{\omega }_1+{\omega }_3]\cup [{\omega }_1+{\omega }_3,{\omega }_3]$ such that $\mathrm{\wp }\left(z_0\right)=0$. … ►The two pairs of edges $[0,{\omega }_1]\cup [{\omega }_1,2{\omega }_3]$ and $[2{\omega }_3,2{\omega }_3-{\omega }_1]\cup [2{\omega }_3-{\omega }_1,0]$ of $R$ are each mapped strictly monotonically by $\mathrm{\wp }$ onto the real line, with $0\to \mathrm{\infty }$, ${\omega }_1\to e_1$, $2{\omega }_3\to -\mathrm{\infty }$; similarly for the other pair of edges. … ►

§23.20(ii) Elliptic Curves

… ►If $a,b\in ℝ$, then $C$ intersects the plane ${ℝ}^2$ in a curve that is connected if $\mathrm{\Delta }\equiv 4a^3+27b^2>0$; if , then the intersection has two components, one of which is a closed loop. …

26: 14.21 Definitions and Basic Properties

§14.21 Definitions and Basic Properties

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14.21.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.1

Referenced by:

§14.21(ii), §14.29, 2nd item

Permalink:

http://dlmf.nist.gov/14.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.21(i), §14.21 and Ch.14

… ►When $z$ is complex $P_{\nu }^{\pm \mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, and ${𝑸}_{\nu }^{\mu }\left(z\right)$ are defined by (14.3.6)–(14.3.10) with $x$ replaced by $z$: the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when $z\in (1,\mathrm{\infty })$, and by continuity elsewhere in the $z$-plane with a cut along the interval $(-\mathrm{\infty },1]$; compare §4.2(i). … ►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. …

27: 1.10 Functions of a Complex Variable

… ►Suppose the subarc $z(t)$, $t\in [t_{j-1},t_j]$ is contained in a domain $D_j$, $j=1,\mathrm{\dots },n$. … ►Let $C$ be a simple closed contour consisting of a segment $\mathrm{𝐴𝐵}$ of the real axis and a contour in the upper half-plane joining the ends of $\mathrm{𝐴𝐵}$. … ►Let $D$ be a domain and $[a,b]$ be a closed finite segment of the real axis. … ►For each $t\in [a,b)$, $f(z,t)$ is analytic in $D$; $f(z,t)$ is a continuous function of both variables when $z\in D$ and $t\in [a,b)$; the integral (1.10.18) converges at $b$, and this convergence is uniform with respect to $z$ in every compact subset $S$ of $D$. … ►

§1.10(xi) Generating Functions

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28: 29.3 Definitions and Basic Properties

§29.3 Definitions and Basic Properties

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§29.3(i) Eigenvalues

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§29.3(iv) Lamé Functions

… ►In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$. … ►To complete the definitions, ${\mathrm{𝐸𝑐}}_{\nu }^m(K,k^2)$ is positive and ${d{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)/dz|}_{z=K}$ is negative. …

29: 25.12 Polylogarithms

… ►The notation ${\mathrm{Li}}_2\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … ►The principal branch has a cut along the interval $[1,\mathrm{\infty })$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i). … ►

25.12.3 $${\mathrm{Li}}_2\left(z\right)+{\mathrm{Li}}_2\left(\frac{z}{z-1}\right)=-\frac{1}{2}{(\ln \left(1-z\right))}^2,$$ $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, ${\mathrm{Li}}_2\left(z\right)$: dilogarithm, $\in$: element of, $\ln z$: principal branch of logarithm function, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable

Keywords:

connection formula

Source:

Maximon (2003, (3.4), p. 2808)

A&S Ref:

27.7.5 (is a modified form with $z=1-x$)

Permalink:

http://dlmf.nist.gov/25.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

… ►For real or complex $s$ and $z$ the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ is defined by … ►The Fermi–Dirac and Bose–Einstein integrals are defined by …

30: 36.13 Kelvin’s Ship-Wave Pattern

… ►When $\rho >1$, that is, everywhere except close to the ship, the integrand oscillates rapidly. … ►Then with the definitions (36.12.12), and the real functions …