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11: 19.14 Reduction of General Elliptic Integrals

… ►If

a_{1}+b_{1}y^{2}=0

, then …If

a_{1}+b_{1}x^{2}=0

, then … ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …

12: Bibliography H

… ►

P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

ⓘ

External Links:: MathReview (C. Tanaka)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

P. I. Hadži (1976b) Integrals that contain a probability function of complicated arguments. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).

ⓘ

External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

P. Henrici (1977) Applied and Computational Complex Analysis. Vol. 2: Special Functions—Integral Transforms—Asymptotics—Continued Fractions. Wiley-Interscience [John Wiley & Sons], New York.

ⓘ

Notes:: Reprinted in 1991.
External Links:: ISBN 0-471-01525-3, MathReview (A. E. Heins), ZentralBlatt
Referenced by:: §1.11(v), Ch.3.
Encodings:: BibTeX

… ►

C. S. Herz (1955) Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.

ⓘ

External Links:: FullText, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §35.1, §35.2, §35.3(ii), §35.5(i), §35.5(ii), §35.5(iii), §35.6(i), §35.6(ii), §35.6(iii), §35.6(iv), §35.7(i), §35.7(ii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), Ch.35, Notations P.
Encodings:: BibTeX

… ►

M. H. Hirata (1975) Flow near the bow of a steadily turning ship. J. Fluid Mech. 71 (2), pp. 283–291.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

…

13: 15.19 Methods of Computation

… ►For

z\in\mathbb{R}

it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval

[0,\frac{1}{2}]

. ►For

z\in\mathbb{C}

it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when

z={\mathrm{e}}^{\pm\pi\mathrm{i}/3}

. This is because the linear transformations map the pair

\{{\mathrm{e}}^{\pi\mathrm{i}/3},{\mathrm{e}}^{-\pi\mathrm{i}/3}\}

onto itself. … ►

§15.19(v) Continued Fractions

►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …

14: 20.11 Generalizations and Analogs

… ►It is a discrete analog of theta functions. If both

m, n

are positive, then

G(m,n)

allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … ►Ramanujan’s theta function

f(a,b)

is defined by … ►In the case

z=0

identities for theta functions become identities in the complex variable

q

, with

\left|q\right|<1

, that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►A further development on the lines of Neville’s notation (§20.1) is as follows. …

15: 19.5 Maclaurin and Related Expansions

… ►

19.5.1

K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.4))
Referenced by:: §19.36(iv), §19.5
Permalink:: http://dlmf.nist.gov/19.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

… ►

19.5.4_1

F\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{% \sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin}% ^{2}\phi\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.8))
Referenced by:: §19.5, Erratum (V1.1.2) for Additions
Permalink:: http://dlmf.nist.gov/19.5.E4_1
Encodings:: pMML, png, TeX
Addition (effective with 1.1.2):: This equation was added.
See also:: Annotations for §19.5 and Ch.19

►

19.5.4_2

E\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}% {\sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},-\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin% }^{2}\phi\right),

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.9))
Referenced by:: §19.5, Erratum (V1.1.2) for Additions
Permalink:: http://dlmf.nist.gov/19.5.E4_2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.2):: This equation was added.
See also:: Annotations for §19.5 and Ch.19

►

19.5.4_3

\Pi\left(\phi,\alpha^{2},k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\sin}^{2m+1}\phi}{(2m+1)m!}{F_{1}}\left(m+\tfrac{1}{2};\tfrac{1}{% 2},1;m+\tfrac{3}{2};{\sin}^{2}\phi,\alpha^{2}{\sin}^{2}\phi\right)k^{2m},

ⓘ

Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Source:: Carlson (1961b, (2.11))
Referenced by:: §19.5, Erratum (V1.1.2) for Additions
Permalink:: http://dlmf.nist.gov/19.5.E4_3
Encodings:: pMML, png, TeX
Addition (effective with 1.1.2):: This equation was added.
See also:: Annotations for §19.5 and Ch.19

… ►Coefficients of terms up to

\lambda^{49}

are given in Lee (1990), along with tables of fractional errors in

K\left(k\right)

and

E\left(k\right)

0.1\leq k^{2}\leq 0.9999

, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). …

16: Bibliography G

… ►

I. Gargantini and P. Henrici (1967) A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

ⓘ

External Links:: FullText, MathReview (D. C. Gilles), ZentralBlatt
Referenced by:: §10.74(v), §3.10(ii).
Encodings:: BibTeX

… ►

A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

… ►

A. Gil and J. Segura (1997) Evaluation of Legendre functions of argument greater than one. Comput. Phys. Comm. 105 (2-3), pp. 273–283.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 18S
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2004a) Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 831; package TOMS), MathReview Entry, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

E. S. Ginsberg and D. Zaborowski (1975) Algorithm 490: The Dilogarithm function of a real argument [S22]. Comm. ACM 18 (4), pp. 200–202.

ⓘ

Notes:: Double-precision Fortran, minimum accuracy 15D. For remark see ACM Trans. Math. Software 2 (1976), p. 112
External Links:: ISSN 0001-0782, Document, Code
Referenced by:: 2nd item.
Encodings:: BibTeX

…

17: Bibliography R

… ►

Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 18D.
External Links:: ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt
Referenced by:: 6th item.
Encodings:: BibTeX

… ►

G. F. Remenets (1973) Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

ⓘ

Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67
External Links:: ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

… ►

S. R. Rengarajan and J. E. Lewis (1980) Mathieu functions of integral orders and real arguments. IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.

ⓘ

Notes:: Includes Fortran program
External Links:: FullText
Referenced by:: §28.36(ii), §28.36(iii).
Encodings:: BibTeX

… ►

D. St. P. Richards (2004) Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.

ⓘ

External Links:: ISSN 0022-4715, Document, MathReview (Stamatis Koumandos), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Wolfgang zu Castell), ZentralBlatt
Referenced by:: §10.23(ii).
Encodings:: BibTeX

…

18: DLMF Project News

error generating summary

19: 19.29 Reduction of General Elliptic Integrals

… ►where the arguments of the

R_{D}

function are, in order,

(a-b)(u-c)

(b-c)(a-u)

(a-b)(b-c)

. … ►The reduction of

I(\mathbf{m})

is carried out by a relation derived from partial fractions and by use of two recurrence relations. …Partial fractions provide a reduction to integrals in which

\mathbf{m}

has at most one nonzero component, and these are then reduced to basic integrals by the recurrence relations. A special case of Carlson (1999, (2.19)) is given by … ►For example, because

t^{3}-a^{3}=(t-a)(t^{2}+at+a^{2})

, we find that when

0\leq a\leq y<x

…

20: 4.21 Identities

… ►

4.21.1

\sin u\pm\cos u=\sqrt{2}\sin\left(u\pm\tfrac{1}{4}\pi\right)=\pm\sqrt{2}\cos% \left(u\mp\tfrac{1}{4}\pi\right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $\sin\NVar{z}$ : sine function
Referenced by:: §4.21(i), Erratum (V1.0.7) for Equation (4.21.1)
Permalink:: http://dlmf.nist.gov/4.21.E1
Encodings:: pMML, png, TeX
Correction (effective with 1.0.7):: Originally the symbol $\pm$ was missing after the second equal sign.
Suggested 2012-09-27 by Dennis M. Heim
See also:: Annotations for §4.21(i), §4.21 and Ch.4

►

4.21.1_5

A\cos u+B\sin u=\sqrt{A^{2}+B^{2}}\cos\left(u-\operatorname{ph}\left(A+B% \mathrm{i}\right)\right),

A,B\in\mathbb{R}

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $\mathbb{R}$ : real line and $\sin\NVar{z}$ : sine function
Referenced by:: §4.21(i), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/4.21.E1_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
Suggested 2018-08-14 by Ted Ersek
See also:: Annotations for §4.21(i), §4.21 and Ch.4

… ►

§4.21(iii) Multiples of the Argument

… ►This result is also valid when

n

is fractional or complex, provided that

-\pi\leq\Re z\leq\pi

. ►

4.21.35

\sin\left(nz\right)=2^{n-1}\prod_{k=0}^{n-1}\sin\left(z+\frac{k\pi}{n}\right),

n=1,2,3,\dots

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $k$ : integer, $n$ : integer and $z$ : complex variable
Referenced by:: §23.10(iii), §4.21(iii)
Permalink:: http://dlmf.nist.gov/4.21.E35
Encodings:: pMML, png, TeX
See also:: Annotations for §4.21(iii), §4.21(iii), §4.21 and Ch.4

…