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1: 26.15 Permutations: Matrix Notation

… ►where the sum is over and $n\ge h>\mathrm{\ell }\ge 1$. … ►A permutation with restricted position specifies a subset $B\subseteq \{1,2,\mathrm{\dots },n\}\times \{1,2,\mathrm{\dots },n\}$. …

2: 26.1 Special Notation

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$x$	real variable.
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$\left(h,k\right)$	greatest common divisor of positive integers $h$ and $k$.

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3: 26.10 Integer Partitions: Other Restrictions

§26.10 Integer Partitions: Other Restrictions

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§26.10(i) Definitions

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§26.10(ii) Generating Functions

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§26.10(iii) Recurrence Relations

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§26.10(v) Limiting Form

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4: 2.1 Definitions and Elementary Properties

… ►(Here and elsewhere in this chapter $\delta$ is an arbitrary small positive constant.) … ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, … ►But for any given set of coefficients $a_0,a_1,a_2,\mathrm{\dots }$, and suitably restricted $𝐗$ there is an infinity of analytic functions $f(x)$ such that (2.1.14) and (2.1.16) apply. For (2.1.14) $𝐗$ can be the positive real axis or any unbounded sector in $ℂ$ of finite angle. …

5: 11.6 Asymptotic Expansions

… ►

11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

►where $\delta$ is an arbitrary small positive constant. …If $\nu$ is real, $z$ is positive, and $m+\frac{1}{2}-\nu \ge 0$, then $R_m(z)$ is of the same sign and numerically less than the first neglected term. … ►

11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(ii), §11.6 and Ch.11

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11.6.9 $${𝐋}_{\nu }\left(\lambda \nu \right)\sim I_{\nu }\left(\lambda \nu \right),$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Referenced by:

§11.6(iii)

Permalink:

http://dlmf.nist.gov/11.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(iii), §11.6 and Ch.11

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6: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

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§26.9(i) Definitions

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§26.9(ii) Generating Functions

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§26.9(iii) Recurrence Relations

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§26.9(iv) Limiting Form

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7: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►When $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed …uniformly with respect to $x\in (0,A]$ in each case, where $A$ is an arbitrary positive constant. ►Other types of approximations when $\kappa \to \mathrm{\infty }$ through positive real values with $\mu$ ($\ge 0$) fixed are as follows. … ►uniformly with respect to $\mu \in [0,(1-\delta )\kappa ]$ and $x\in (0,(1-\delta )(2\kappa +2\sqrt{{\kappa }^2-{\mu }^2})]$, where $\delta$ again denotes an arbitrary small positive constant. … ►This expansion is simpler in form than the expansions of Dunster (1989) that correspond to the approximations given in §13.21(iii), but the conditions on $\mu$ are more restrictive. …

8: 1.10 Functions of a Complex Variable

… ►and the integration contour is described once in the positive sense. … ►Here and elsewhere in this subsection the path $C$ is described in the positive sense. … ►where $N$ and $P$ are respectively the numbers of zeros and poles, counting multiplicity, of $f$ within $C$, and ${\mathrm{\Delta }}_C(\mathrm{ph}f(z))$ is the change in any continuous branch of $\mathrm{ph}\left(f(z)\right)$ as $z$ passes once around $C$ in the positive sense. … ►(a) By introducing appropriate cuts from the branch points and restricting $F(z)$ to be single-valued in the cut plane (or domain). … ►The last condition means that given $ϵ$ ($>0$) there exists a number $a_0\in [a,b)$ that is independent of $z$ and is such that …

9: 15.12 Asymptotic Approximations

… ►Let $\delta$ denote an arbitrary small positive constant. … ►

$\mathrm{\Re }z>\frac{1}{2}$ and ${\alpha }_{-}-\frac{1}{2}\pi +\delta \le \mathrm{ph}c\le {\alpha }_{+}+\frac{1}{2}\pi -\delta$, where

15.12.1 $${\alpha }_{\pm }=\arctan \left(\frac{\mathrm{ph}z-\mathrm{ph}\left(1-z\right)\mp \pi }{\ln |1-z^{-1}|}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable and ${\alpha }_{\pm }$

Permalink:

http://dlmf.nist.gov/15.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(ii), §15.12 and Ch.15

with $z$ restricted so that $\pm {\alpha }_{\pm }\in [0,\frac{1}{2}\pi )$.

… ►Again, throughout this subsection $\delta$ denotes an arbitrary small positive constant, and $a,b,c,z$ are real or complex and fixed. …

10: Mathematical Introduction

… ►This is because $𝐅$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $𝐅$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. … ► ►►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
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$mod$ or modulo	$m\equiv n(modp)$ means $p$ divides $m-n$, where $m$, $n$, and $p$ are positive integers with $m>n$.
$\mathbb{N}$	set of all positive integers.
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