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1: 19.2 Definitions

… ►The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$. … ►When $x$ and $y$ are positive, $R_C(x,y)$ is an inverse circular function if and an inverse hyperbolic function (or logarithm) if $x>y$: …For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). …

2: 34.6 Definition: $\mathit{9}j$ Symbol

§34.6 Definition: $\mathit{9}j$ Symbol

►The $\mathit{9}j$ symbol may be defined either in terms of $\mathit{3}j$ symbols or equivalently in terms of $\mathit{6}j$ symbols: ►

34.6.1

ⓘ

Defines:

: $\mathit{9}j$ symbol

Symbols:

: $\mathit{3}j$ symbol, $j,j_r$: non-negative integers or non-negative integers plus one half., $r$: nonnegative integer and $s$: integer

Permalink:

http://dlmf.nist.gov/34.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §34.6 and Ch.34

►

34.6.2

ⓘ

Symbols:

: $\mathit{9}j$ symbol, : $\mathit{6}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Referenced by:

§34.13

Permalink:

http://dlmf.nist.gov/34.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §34.6 and Ch.34

►The $\mathit{9}j$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

3: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…

4: 26.5 Lattice Paths: Catalan Numbers

… ► $C\left(n\right)$ is the Catalan number. …(Sixty-six equivalent definitions of $C\left(n\right)$ are given in Stanley (1999, pp. 219–229).) … ►

Table 26.5.1: Catalan numbers.

► ►►►

$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
…
1	1	8	1430	15	96 94845
…

►

… ►

26.5.3 $$C\left(n+1\right)=\sum _{k=0}^{n}C\left(k\right)C\left(n-k\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

►

26.5.4 $$C\left(n+1\right)=\frac{2(2n+1)}{n+2}C\left(n\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iii), §26.5 and Ch.26

…

5: Bibliography H

… ►

Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

ⓘ

Notes:

A reprinting with corrections appeared in 2000.

External Links:

ISBN 3-540-56670-8, MathReview, ZentralBlatt

Referenced by:

§3.7(v), §3.7(i).

Encodings:

BibTeX

… ►

Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

ⓘ

External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.16.

Encodings:

BibTeX

… ►

Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

ⓘ

External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

BibTeX

… ►

Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

ⓘ

External Links:

Document

Referenced by:

§33.7.

Encodings:

BibTeX

… ►

On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§24.15(iii).

Encodings:

BibTeX

…

6: 1.12 Continued Fractions

… ► $C_n$ is called the $n$th approximant or convergent to $C$ . … ►Define … ►A contraction of a continued fraction $C$ is a continued fraction $C^{\prime }$ whose convergents $\{C_n^{\prime }\}$ form a subsequence of the convergents $\{C_n\}$ of $C$. Conversely, $C$ is called an extension of $C^{\prime }$. … ►Then the convergents $C_n$ satisfy …

7: 33.20 Expansions for Small $|ϵ|$

… ►where ►

33.20.4 $${𝖥}_k(\mathrm{\ell };r)=\sum _{p=2k}^{3k}{(2r)}^{(p+1)/2}{C}_{k,p}{J}_{2\mathrm{\ell }+1+p}\left(\sqrt{8r}\right),$$ $r>0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, ${𝖥}_k(\mathrm{\ell };r)$: coefficient and $C_{k,p}$: coefficients

Permalink:

http://dlmf.nist.gov/33.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.20(ii), §33.20 and Ch.33

… ►The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$, $C_{1,0}=0$, and … ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and …The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). …

8: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ►

Table 26.6.2: Motzkin numbers $M(n)$.

► ►►►

$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
0	1	4	9	8	323	12	15511	16	8 53467
…

►

… ►

Table 26.6.3: Narayana numbers $N(n,k)$.

► ►►►

$n$	$k$
	…
8	0	1	28	196	490	490	196	28	1
…

►

… ►

Table 26.6.4: Schröder numbers $r(n)$.

► ►►►

$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
$0$	$1$	$4$	$90$	$8$	$41586$	$12$	$\mathrm{272\hspace{0.33em}97738}$	$16$	$\mathrm{2\hspace{0.33em}09271\hspace{0.33em}56706}$
…

►

…

9: 7.3 Graphics

… ►

►►

Figure 7.3.2: Dawson’s integral $F\left(x\right)$, $-3.5\le x\le 3.5$. Magnify

►

►►

Figure 7.3.3: Fresnel integrals $C\left(x\right)$ and $S\left(x\right)$, $0\le x\le 4$. Magnify

►

►►

Figure 7.3.4: ${|ℱ\left(x\right)|}^2$, $-8\le x\le 8$. … Magnify

…

10: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form … ►Given $P$, calculate $2P$, $4P$, $8P$ by doubling as above. …If any of $2P$, $4P$, $8P$ is not an integer, then the point has infinite order. Otherwise observe any equalities between $P$, $2P$, $4P$, $8P$, and their negatives. The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from $2P=-P$, $4P=-P$, $4P=-2P$, $8P=P$, $8P=-P$, $8P=-2P$, or $8P=-4P$. …