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21: 29.3 Definitions and Basic Properties

… ►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 $[0,2K)$. ►

Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
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22: 32.14 Combinatorics

… ►Let $S_N$ be the group of permutations $𝝅$ of the numbers $1,2,\mathrm{\dots },N$ (§26.2). …and $w(x)$ satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$ and boundary conditions …

23: 1.5 Calculus of Two or More Variables

… ►Sufficient conditions for validity are: (a) $f$ and $\partial f/\partial x$ are continuous on a rectangle $a\le x\le b$, $c\le y\le d$; (b) when $x\in [a,b]$ both $\alpha (x)$ and $\beta (x)$ are continuously differentiable and lie in $[c,d]$. … ►Suppose also that ${\int }_c^{d}f(x,y)dy$ converges and ${\int }_c^d(\partial f/\partial x)dy$ converges uniformly on $a\le x\le b$, that is, given any positive number $ϵ$, however small, we can find a number $c_0\in [c,d)$ that is independent of $x$ and is such that ►

1.5.23

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $ϵ$: positive number and $\left|x\right|$: absolute value of $x$

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http://dlmf.nist.gov/1.5.E23

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See also:

Annotations for §1.5(iv), §1.5(iv), §1.5 and Ch.1

… ►Sufficient conditions for the limit to exist are that $f(x,y)$ is continuous, or piecewise continuous, on $R$. …

24: 3.9 Acceleration of Convergence

… ►All sequences (series) in this section are sequences (series) of real or complex numbers. … ►In Weniger’s transformations the numbers $c_{j,k,n}$ in (3.9.13) are chosen as follows: …where ${\left(a\right)}_0=1$ and ${\left(a\right)}_j=a(a+1)\mathrm{\cdots }(a+j-1)$ are Pochhammer symbols (§5.2(iii)), and the constants $\beta$ and $\gamma$ are chosen arbitrarily subject to certain conditions. …

25: Errata

… ►This release increments the minor version number and contains considerable additions of new material and clarifications. … ►These additions were facilitated by an extension of the scheme for reference numbers; with “_” introducing intermediate numbers. … ►

There were clarifications made in the conditions on the parameter $a$ in $U(a,b,z)$ of those equations.

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The condition for the validity of (8.17.5) is that $m$ and $n$ are positive integers and . Previously, no conditions were stated.

Reported 2011-03-23 by Stephen Bourn.

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Originally it was implied that $R_C(x,y)$ is an elliptic integral. It was clarified that $R_{-a}(𝐛;𝐳)$ is an elliptic integral iff the stated conditions hold; originally these conditions were stated as sufficient but not necessary. In particular, $R_C(x,y)$ does not satisfy these conditions.

Reported 2010-11-23.

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26: 25.14 Lerch’s Transcendent

… ►If $s$ is not an integer then ; if $s$ is a positive integer then $a\ne 0,-1,-2,\mathrm{\dots }$; if $s$ is a non-positive integer then $a$ can be any complex number. … ►With the conditions of (25.14.1) and $m=1,2,3,\mathrm{\dots }$, …

27: 19.20 Special Cases

… ►In this subsection, and also §§19.20(ii)–19.20(v), the variables of all $R$-functions satisfy the constraints specified in §19.16(i) unless other conditions are stated. … ► Schneider that this is a transcendental number. … ► Schneider that this is a transcendental number. …

28: 19.29 Reduction of General Elliptic Integrals

… ►and $\alpha ,\beta ,\gamma ,\delta$ is any permutation of the numbers $1,2,3,4$, then … ►Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. … ►Define also $\mathrm{𝟎}=(0,\mathrm{\dots },0)$ and retain the notation and conditions associated with (19.29.1) and (19.29.2). … ►where $\alpha ,\beta ,\gamma$ is any permutation of the numbers $1,2,3$, and …

29: 1.9 Calculus of a Complex Variable

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§1.9(i) Complex Numbers

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Polar Representation

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Modulus and Phase

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Powers

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Winding Number

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30: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Self-adjoint extensions of (1.18.28) and the Weyl alternative

… ► A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …Then, if the linear form $ℬ$ is nonzero, the condition $ℬ(f)=0$ is called a boundary condition at $a$. Boundary values and boundary conditions for the end point $b$ are defined in a similar way. … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. …