%E6%BE%B3%E9%97%A8%E8%B5%8C%E5%9C%BA%E5%AE%98%E7%BD%91,%E6%BE%B3%E9%97%A8%E8%B5%8C%E5%9C%BA%E6%8E%92%E5%90%8D,%E6%BE%B3%E9%97%A8%E8%B5%8C%E5%9C%BA%E6%94%BB%E7%95%A5,%E3%80%90%E6%BE%B3%E9%97%A8%E8%B5%8C%E5%9C%BA%E7%BD%91%E5%9D%80%E2%88%B6789yule.com%E3%80%91%E6%BE%B3%E9%97%A8%E7%BD%91%E4%B8%8A%E8%B5%8C%E5%9C%BA%E6%B8%B8%E6%88%8F%E7%A7%8D%E7%B1%BB,%E6%BE%B3%E9%97%A8%E8%B5%8C%E5%9C%BA%E4%BB%8B%E7%BB%8D,%E6%BE%B3%E9%97%A8%E8%B5%8C%E5%9C%BA%E7%8E%A9%E6%B3%95,%E8%B5%8C%E5%9C%BA%E6%B8%B8%E6%88%8F%E8%A7%84%E5%88%99,%E6%BE%B3%E9%97%A8%E8%B5%8C%E5%9C%BAapp,%E3%80%90%E5%8D%9A%E5%BD%A9%E7%BD%91%E5%9D%80%E2%88%B6789yule.com%E3%80%91

SearchAdvancedHelp

(0.034 seconds)

11—20 of 760 matching pages

11: 26.13 Permutations: Cycle Notation

… ►An explicit representation of $\sigma$ can be given by the $2\times n$ matrix: … ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►A permutation that consists of a single cycle of length $k$ can be written as the composition of $k-1$ two-cycles (read from right to left): …A permutation with cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$ can be written as a product of $a_2+2a_3+\mathrm{\cdots }+(n-1)a_n=n-(a_1+a_2+\mathrm{\cdots }+a_n)$ transpositions, and no fewer. …

12: DLMF Project News

error generating summary

13: 33.13 Complex Variable and Parameters

… ►The functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, and $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ may be extended to noninteger values of $\mathrm{\ell }$ by generalizing $(2\mathrm{\ell }+1)!=\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right)$, and supplementing (33.6.5) by a formula derived from (33.2.8) with $U(a,b,z)$ expanded via (13.2.42). ►These functions may also be continued analytically to complex values of $\rho$, $\eta$, and $\mathrm{\ell }$. The quantities $C_{\mathrm{\ell }}\left(\eta \right)$, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$, and $R_{\mathrm{\ell }}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that ►

33.13.1 $$C_{\mathrm{\ell }}\left(\eta \right)=2^{\mathrm{\ell }}{\mathrm{e}}^{\mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)-(\pi \eta /2)}\mathrm{\Gamma }\left(\mathrm{\ell }+1-\mathrm{i}\eta \right)/\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right),$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Permalink:

http://dlmf.nist.gov/33.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

… ►

33.13.2 $$R_{\mathrm{\ell }}=(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)/C_{\mathrm{\ell }-1}\left(\eta \right).$$

ⓘ

Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $R_{\mathrm{\ell }}$: factor

Permalink:

http://dlmf.nist.gov/33.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

…

14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►In the following let $V$ be a Hilbert space. … ►and thus $T$ is said to be formally self adjoint. … ►In what follows $T$ will be taken to be a self adjoint extension of $ℒ$ following the discussion ending the prior sub-section. … ►These eigenvalues will be assumed distinct, i. … ►Should $q(x)$ be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). …

15: 19.36 Methods of Computation

… ►Polynomials of still higher degree can be obtained from (19.19.5) and (19.19.7). … ►Alternatively, the first duplication is done analytically as in Carlson and FitzSimons (2000), where further information can be found. … ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). …The cancellations can be eliminated, however, by using (19.25.10). … ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …

16: 10.36 Other Differential Equations

… ►The quantity ${\lambda }^2$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-{\lambda }^2$ if at the same time the symbol $𝒞$ in the given solutions is replaced by $𝒵$. … ►Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $\mathrm{i}z$.

17: 34.9 Graphical Method

§34.9 Graphical Method

… ►Thus, any analytic expression in the theory, for example equations (34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be represented by a diagram; conversely, any diagram represents an analytic equation. …For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

18: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors. … ►However, $w_n$ can be computed successfully in these circumstances by boundary-value methods, as follows. … ►The least value of $N$ that satisfies (3.6.9) is found to be 16. … ►Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. …

19: 8.23 Statistical Applications

… ►The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

20: Ian J. Thompson

Profile
Ian J. Thompson

… ►Ian J. Thompson (b. …