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1: 26.13 Permutations: Cycle Notation

… ►In cycle notation, the elements in each cycle are put inside parentheses, ordered so that

\sigma(j)

immediately follows

j

or, if

j

is the last listed element of the cycle, then

\sigma(j)

is the first element of the cycle. … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by

{\left(1,3,2,5,7\right)}{\left(6,8\right)}={\left(2,3\right)}\*{\left(1,2% \right)}\*{\left(4,5\right)}{\left(3,4\right)}{\left(2,3\right)}{\left(3,4% \right)}{\left(4,5\right)}{\left(6,7\right)}{\left(5,6\right)}{\left(7,8\right% )}\*{\left(6,7\right)}

\mathop{\mathrm{inv}}({\left(1,3,2,5,7\right)}{\left(6,8\right)})=11

3: Bibliography F

… ►

B. R. Fabijonas (2004) Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

ⓘ

Notes:: Single- and double-precision Fortran, maximum accuracy 32S.
External Links:: ISSN 0098-3500, Document, Code, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, 2nd item, 1st item.
Encodings:: BibTeX

… ►

H. E. Fettis and J. C. Caslin (1969) A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp(i\theta)$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.
External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(ii), §19.37(ii).
Encodings:: BibTeX

… ►

N. Fleury and A. Turbiner (1994) Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt
Referenced by:: §13.3(ii), §15.5(i), §16.3(i).
Encodings:: BibTeX

… ►

A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.13(i), §32.7(i), §32.7(iii).
Encodings:: BibTeX

… ►

L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

ⓘ

Notes:: Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.
External Links:: ISSN 0001-0782, Document, Remark
Referenced by:: §8.28(ii).
Encodings:: BibTeX

…

4: Guide to Searching the DLMF

… ►Be careful, however, because if you put quotes around math expressions involving math symbols, you may not get the matches you’d expect. …

5: 19.22 Quadratic Transformations

… ►The AGM,

M\left(a_{0},g_{0}\right)

, of two positive numbers

a_{0}

and

g_{0}

is defined in §19.8(i). …where …where

p_{0}>0

and …(If

p_{0}=a_{0}

, then

p_{n}=a_{n}

and (19.22.13) reduces to (19.22.11).) … ►and the corresponding equations with

z

z_{+}

, and

z_{-}

replaced by

p

p_{+}

, and

p_{-}

, respectively. …

6: 19.17 Graphics

… ►To view

R_{F}\left(0,y,1\right)

and

2R_{G}\left(0,y,1\right)

for complex

y

, put

y=1-k^{2}

, use (19.25.1), and see Figures 19.3.7–19.3.12. … ►To view

R_{F}\left(0,y,1\right)

and

2R_{G}\left(0,y,1\right)

for complex

y

, put

y=1-k^{2}

, use (19.25.1), and see Figures 19.3.7–19.3.12. …

7: 19.28 Integrals of Elliptic Integrals

… ►

19.28.4

\int_{0}^{1}t^{\sigma-1}(1-t)^{c-1}R_{-a}\left(b_{1},b_{2};t,1\right)\,\mathrm% {d}t=\frac{\Gamma\left(c\right)\Gamma\left(\sigma\right)\Gamma\left(\sigma+b_{% 2}-a\right)}{\Gamma\left(\sigma+c-a\right)\Gamma\left(\sigma+b_{2}\right)},

c=b_{1}+b_{2}>0

\Re\sigma>\max(0,a-b_{2})

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\sigma$ : parameter
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

… ►

19.28.5

\int_{z}^{\infty}R_{D}\left(x,y,t\right)\,\mathrm{d}t=6R_{F}\left(x,y,z\right),

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

►

19.28.6

\int_{0}^{1}R_{D}\left(x,y,v^{2}z+(1-v^{2})p\right)\,\mathrm{d}v=R_{J}\left(x,% y,z,p\right).

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.28, §19.28
Permalink:: http://dlmf.nist.gov/19.28.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

… ►

19.28.8

\int_{0}^{\infty}R_{J}\left(tx,y,z,tp\right)\,\mathrm{d}t=\frac{6}{\sqrt{p}}R_% {C}\left(p,x\right)R_{F}\left(0,y,z\right).

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Referenced by:: §19.28
Permalink:: http://dlmf.nist.gov/19.28.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.28 and Ch.19

… ►To replace a single component of

\mathbf{z}

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

by several different variables (as in (19.28.6)), see Carlson (1963, (7.9)). …

8: 19.20 Special Cases

… ►

§19.20(i) $R_{F}\left(x,y,z\right)$

… ►

§19.20(ii) $R_{G}\left(x,y,z\right)$

… ►

§19.20(v) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

►Define

c=\sum_{j=1}^{n}b_{j}

. …where

T_{N}

is defined by (19.19.1). …

9: 19.26 Addition Theorems

… ►An equivalent version for

R_{C}

is … ►

19.26.20

R_{D}\left(x,y,z\right)=2R_{D}\left(x+\lambda,y+\lambda,z+\lambda\right)+\frac% {3}{\sqrt{z}(z+\lambda)}.

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

►

19.26.21

2R_{G}\left(x,y,z\right)=4R_{G}\left(x+\lambda,y+\lambda,z+\lambda\right)-% \lambda R_{F}\left(x,y,z\right)-\sqrt{x}-\sqrt{y}-\sqrt{z}.

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.36(i)
Permalink:: http://dlmf.nist.gov/19.26.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

►

19.26.22

R_{J}\left(x,y,z,p\right)=2R_{J}\left(x+\lambda,y+\lambda,z+\lambda,p+\lambda% \right)+3R_{C}\left(\alpha^{2},\beta^{2}\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\lambda$ : positive, $\alpha$ , $\beta$ , $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

… ►

19.26.25

R_{C}\left(x,y\right)=2R_{C}\left(x+\lambda,y+\lambda\right),

\lambda=y+2\sqrt{x}\sqrt{y}

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$ : positive, $x$ : positive and $y$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

…

10: 19.3 Graphics

… ►

See accompanying text — Figure 19.3.3: $F\left(\phi,k\right)$ as a function of $k^{2}$ and ${\sin}^{2}\phi$ for $-1\leq k^{2}\leq 2$ , $0\leq{\sin}^{2}\phi\leq 1$ . …If ${\sin}^{2}\phi=1/k^{2}$ ( $<1$ ), then it has the value $K\left(1/k\right)/k$ : put $c=k^{2}$ in (19.25.5) and use (19.25.1). Magnify 3D Help

►

…