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11: DLMF Project News

error generating summary

12: 18.2 General Orthogonal Polynomials

… ►The moments for an orthogonality measure $d\mu (x)$ are the numbers … ►are the Christoffel numbers, see also (3.5.18). … ►Nevai (1979, p.39) defined the class $𝒮$ of orthogonality measures with support inside $[-1,1]$ such that the absolutely continuous part $w(x)dx$ has $w$ in the Szegő class $𝒢$. … ►If $d\mu \in 𝐌(a,b)$ then the interval $[b-a,b+a]$ is included in the support of $d\mu$, and outside $[b-a,b+a]$ the measure $d\mu$ only has discrete mass points $x_k$ such that $b\pm a$ are the only possible limit points of the sequence $\{x_k\}$, see Máté et al. (1991, Theorem 10). … ►for $x,y$ in the support of the orthogonality measure and $z$ such that the series in (18.2.41) converges absolutely for all these $x,y$. …

13: 6.4 Analytic Continuation

… ►

6.4.2 $$E_1\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)=E_1\left(z\right)-2m\pi \mathrm{i},$$ $m\in \mathbb{Z}$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers and $z$: complex variable

Permalink:

http://dlmf.nist.gov/6.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

…

14: 35.2 Laplace Transform

… ►Suppose there exists a constant ${𝐗}_0\in 𝛀$ such that for all $𝐗\in 𝛀$. Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. … ►Assume that ${\int }_{𝓢}\left|g(𝐔+\mathrm{i}𝐕)\right|d𝐕$ converges, and also that its limit as $𝐔\to \mathrm{\infty }$ is $0$. …where the integral is taken over all $𝐙=𝐔+\mathrm{i}𝐕$ such that $𝐔>{𝐗}_0$ and $𝐕$ ranges over $𝓢$. … ►If $g_j$ is the Laplace transform of $f_j$, $j=1,2$, then $g_1{g}_2$ is the Laplace transform of the convolution $f_1\ast f_2$, where …

15: 9.18 Tables

… ►

Rothman (1954b) tabulates ${\int }_0^x\mathrm{Ai}\left(t\right)dt$ and ${\int }_0^x\mathrm{Bi}\left(t\right)dt$ for $x=-10(.1)\mathrm{\infty }$ and $-10(.1)2$, respectively; 7D. The entries in the columns headed ${\int }_0^x\mathrm{Ai}\left(-x\right)dx$ and ${\int }_0^x\mathrm{Bi}\left(-x\right)dx$ all have the wrong sign. The tables are reproduced in Abramowitz and Stegun (1964, Chapter 10), and the sign errors are corrected in later reprintings.

…

16: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …

17: 29.14 Orthogonality

… ►

29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

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See also:

Annotations for §29.14 and Ch.29

►

29.14.6 $${\mathrm{𝑑𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑑𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E6

Encodings:

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See also:

Annotations for §29.14 and Ch.29

… ►In each system $n$ ranges over all nonnegative integers and $m=0,1,\mathrm{\dots },n$. When combined, all eight systems (29.14.1) and (29.14.4)–(29.14.10) form an orthogonal and complete system with respect to the inner product …

18: 8.19 Generalized Exponential Integral

… ►For $j=1,2,3,\mathrm{\dots }$, ►

8.19.15 $$\frac{{\partial }^j{E}_p\left(z\right)}{{\partial p}^j}={(-1)}^j{\int }_1^{\mathrm{\infty }}{(\ln t)}^j{t}^{-p}{\mathrm{e}}^{-zt}dt,$$ $\mathrm{\Re }z>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\mathrm{\Re }$: real part, $z$: complex variable, $p$: parameter and $j$: numbers

Permalink:

http://dlmf.nist.gov/8.19.E15

Encodings:

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See also:

Annotations for §8.19(v), §8.19(v), §8.19 and Ch.8

… ►

8.19.18 $$E_p\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)=\frac{2\pi \mathrm{i}{\mathrm{e}}^{mp\pi \mathrm{i}}}{\mathrm{\Gamma }\left(p\right)}\frac{\sin \left(mp\pi \right)}{\sin \left(p\pi \right)}{z}^{p-1}+E_p\left(z\right),$$ $m\in \mathbb{Z}$, $z\ne 0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\sin z$: sine function, $z$: complex variable and $p$: parameter

Referenced by:

§2.11(ii)

Permalink:

http://dlmf.nist.gov/8.19.E18

Encodings:

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See also:

Annotations for §8.19(viii), §8.19 and Ch.8

…

19: 1.4 Calculus of One Variable

… ►If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. … ►Similarly, assume that ${\int }_{-b}^{b}f(x)dx$ exists for all finite values of $b$ ($>0$), but not necessarily when $b=\mathrm{\infty }$. … ►Lastly, whether or not the real numbers $a$ and $b$ satisfy , and whether or not they are finite, we define ${𝒱}_{a,b}\left(f\right)$ by (1.4.34) whenever this integral exists. …

20: 2.5 Mellin Transform Methods

… ►Let $f(t)$ be a locally integrable function on $(0,\mathrm{\infty })$, that is, ${\int }_{\rho }^{T}f(t)dt$ exists for all $\rho$ and $T$ satisfying . … ►Now suppose that there is a real number $p_{jk}$ in $D_{jk}$ such that the Parseval formula (2.5.5) applies and …If, in addition, there exists a number $q_{jk}>p_{jk}$ such that ►

2.5.34 $$\underset{p_{jk}\le x\le q_{jk}}{sup}\left|G_{jk}(x+\mathrm{i}y)\right|\to 0,$$ $y\to \pm \mathrm{\infty }$,

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $sup$: least upper bound (supremum), $G_{jk}(z)$: function, $p_{jk}$: real number and $q_{jk}>p_{jk}$: real number

Permalink:

http://dlmf.nist.gov/2.5.E34

Encodings:

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See also:

Annotations for §2.5(ii), §2.5 and Ch.2

… ►Since $ℳ{\mathrm{e}}^{-t}\left(z\right)=\mathrm{\Gamma }\left(z\right)$, by the Parseval formula (2.5.5), there are real numbers $p_1$ and $p_2$ such that , , and …