高仿花旗银行报表【言正 微aptao168】45S

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11: 33.4 Recurrence Relations and Derivatives

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$S_{\mathrm{\ell }}={\displaystyle \frac{\mathrm{\ell }}{\rho }}+{\displaystyle \frac{\eta }{\mathrm{\ell }}},$

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$T_{\mathrm{\ell }}=S_{\mathrm{\ell }}+S_{\mathrm{\ell }+1}.$

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33.4.3 $$X_{\mathrm{\ell }}^{\prime }=R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-S_{\mathrm{\ell }}{X}_{\mathrm{\ell }},$$ $\mathrm{\ell }\ge 1$,

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Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.1

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

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33.4.4 $$X_{\mathrm{\ell }}^{\prime }=S_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }}-R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1},$$ $\mathrm{\ell }\ge 0$.

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Symbols:

$\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: any Coulomb function

A&S Ref:

14.2.2

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.4 and Ch.33

12: 30.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. … ►Flammer (1957) and Abramowitz and Stegun (1964) use ${\lambda }_{mn}(\gamma )$ for ${\lambda }_n^m\left({\gamma }^2\right)+{\gamma }^2$, $R_{mn}^{(j)}(\gamma ,z)$ for $S_n^{m(j)}(z,\gamma )$, and ►

$S_{mn}^{(1)}(\gamma ,x)=d_{mn}(\gamma ){\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$

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$S_{mn}^{(2)}(\gamma ,x)=d_{mn}(\gamma ){\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2),$

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$S_{mn}^{(1)}(\gamma ,0)={(-1)}^m{𝖯}_n^m\left(0\right),$ $n-m$ even,

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13: 30.17 Tables

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Hanish et al. (1970) gives ${\lambda }_n^m\left({\gamma }^2\right)$ and $S_n^{m(j)}(z,\gamma )$, $j=1,2$, and their first derivatives, for $0\le m\le 2$, $m\le n\le m+49$, $-1600\le {\gamma }^2\le 1600$. The range of $z$ is given by $1\le z\le 10$ if ${\gamma }^2>0$, or $z=-\mathrm{i}\xi$, $0\le \xi \le 2$ if . Precision is 18S.

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EraŠevskaja et al. (1973, 1976) gives $S^{m(j)}(\mathrm{i}y,-\mathrm{i}c)$, $S^{m(j)}(z,\gamma )$ and their first derivatives for $j=1,2$, $0.5\le c\le 8$, $y=0,0.5,1,1.5$, $0.5\le \gamma \le 8$, $z=1.01,1.1,1.4,1.8$. Precision is 15S.

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14: 5.19 Mathematical Applications

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$S={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{a}_k,$

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5.19.3 $$S=\psi \left(\frac{1}{2}\right)-2\psi \left(\frac{2}{3}\right)-\gamma =3\ln 3-2\ln 2-\frac{1}{3}\pi \sqrt{3}.$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/5.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

… ►The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by … ►

$S={\displaystyle \frac{2{\pi }^{\frac{1}{2}n}{r}^{n-1}}{\mathrm{\Gamma }\left(\frac{1}{2}n\right)}}={\displaystyle \frac{n}{r}}V.$

15: 7.1 Special Notation

… ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. …

16: 35.5 Bessel Functions of Matrix Argument

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35.5.2 $$A_{\nu }\left(𝐓\right)=A_{\nu }\left(𝟎\right)\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\sum _{|\kappa |=k}\frac{1}{{\left[\nu +\frac{1}{2}(m+1)\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $\nu \in ℂ$, $𝐓\in 𝓢$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $k$: nonnegative integer, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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35.5.4 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\nu }{A}_{\nu }\left(𝐒𝐗\right)d𝐗=\mathrm{etr}\left(-𝐒{𝐓}^{-1}\right){\left|𝐓\right|}^{-\nu -\frac{1}{2}(m+1)},$$ $𝐒\in 𝓢$, $𝐓\in 𝛀$; $\mathrm{\Re }\left(\nu \right)>-1$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $\mathrm{\Re }$: real part, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.5 ${\nu }_j\in ℂ$, $\mathrm{\Re }\left({\nu }_j\right)>-1$, $j=1,2$; ${𝐒}_1,{𝐒}_2\in 𝓢$; $𝐓\in 𝛀$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, : less-than for matrices, $j$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.7 $${\int }_{𝛀}{A}_{{\nu }_1}\left(𝐓𝐗\right)B_{-{\nu }_2}\left(𝐒𝐗\right){\left|𝐗\right|}^{{\nu }_1}d𝐗=\frac{1}{A_{{\nu }_1+{\nu }_2}\left(𝟎\right)}{\left|𝐒\right|}^{{\nu }_2}{\left|𝐓+𝐒\right|}^{-({\nu }_1+{\nu }_2+\frac{1}{2}(m+1))},$$ $\mathrm{\Re }\left({\nu }_1+{\nu }_2\right)>-1$; $𝐒,𝐓\in 𝛀$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind), $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.8 $${\int }_{𝐎(m)}\mathrm{etr}\left(𝐒𝐇\right)\mathrm{d}𝐇=\frac{A_{-1/2}\left(-\frac{1}{4}𝐒{𝐒}^{\mathrm{T}}\right)}{A_{-1/2}\left(𝟎\right)},$$ $𝐒$ arbitrary.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐒$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.10, §35.9

Permalink:

http://dlmf.nist.gov/35.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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17: 26.13 Permutations: Cycle Notation

… ► ${𝔖}_n$ denotes the set of permutations of $\{1,2,\mathrm{\dots },n\}$. $\sigma \in {𝔖}_n$ is a one-to-one and onto mapping from $\{1,2,\mathrm{\dots },n\}$ to itself. … ►The number of elements of ${𝔖}_n$ with cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$ is given by (26.4.7). … ►The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: … ►Given a permutation $\sigma \in {𝔖}_n$, the inversion number of $\sigma$, denoted $inv(\sigma )$, is the least number of adjacent transpositions required to represent $\sigma$. …

18: 7.4 Symmetry

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$S\left(-z\right)=-S\left(z\right),$

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$S\left(\mathrm{i}z\right)=-\mathrm{i}S\left(z\right).$

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19: 26.18 Counting Techniques

… ►Let $A_1,A_2,\mathrm{\dots },A_n$ be subsets of a set $S$ that are not necessarily disjoint. Then the number of elements in the set $S\setminus (A_1\cup A_2\cup \mathrm{\cdots }\cup A_n)$ is ►

26.18.1

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Symbols:

$\left|A\right|$: number of elements of a finite set $A$ , $\cap$: intersection, $\setminus$: set subtraction, $\cup$: union, $j$: nonnegative integer, $n$: nonnegative integer, $A_k$: subsets and $S$: set

Permalink:

http://dlmf.nist.gov/26.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18 and Ch.26

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20: 1.6 Vectors and Vector-Valued Functions

… ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►A parametrized surface $S$ is defined by … ►The area $A(S)$ of a parametrized smooth surface is given by … ►The integral of a continuous function $f(x,y,z)$ over a surface $S$ is … ►Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. …