relation to sine and cosine integrals

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21: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►In (19.14.1)–(19.14.3) both the integrand and

\cos\phi

are assumed to be nonnegative. … ►The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). …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. …

22: Bibliography H

… ►

L. Habsieger (1988) Une

q

-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §17.13.
Encodings:: BibTeX

… ►

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 (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

ⓘ

External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

H. Hancock (1958) Elliptic Integrals. Dover Publications Inc., New York.

ⓘ

Notes:: Unaltered reprint of original edition published by Wiley, New York, 1917.
External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §19.8(ii).
Encodings:: BibTeX

… ►

F. E. Harris (2000) Spherical Bessel expansions of sine, cosine, and exponential integrals. Appl. Numer. Math. 34 (1), pp. 95–98.

ⓘ

External Links:: ISSN 0168-9274, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii), §6.10(ii), §6.15.
Encodings:: BibTeX

…

23: Bibliography B

… ►

G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

ⓘ

External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: 1st item.
Encodings:: BibTeX

►

G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

ⓘ

External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: 2nd item, 1st item.
Encodings:: BibTeX

… ►

V. Britanak, P. C. Yip, and K. R. Rao (2007) Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.

ⓘ

External Links:: ISBN 978-0-12-373624-6; 0-12-373624-2, MathReview (Gerd Teschke)
Referenced by:: §18.3.
Encodings:: BibTeX

… ►

R. Bulirsch (1967) Numerical calculation of the sine, cosine and Fresnel integrals. Numer. Math. 9 (5), pp. 380–385.

ⓘ

Notes:: Algol 60 procedures for sine, cosine, and 2 Fresnel integrals are included.
External Links:: ISSN 0029-599X, ISSN 0029-599X, Document, MathReview Entry, ZentralBlatt
Referenced by:: 2nd item, §7.25(iv).
Encodings:: BibTeX

… ►

P. J. Bushell (1987) On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

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External Links:: ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.13(i).
Encodings:: BibTeX

…

24: 19.2 Definitions

… ►

§19.2(i) General Elliptic Integrals

… ►

§19.2(ii) Legendre’s Integrals

… ►

§19.2(iii) Bulirsch’s Integrals

… ►Lastly, corresponding to Legendre’s incomplete integral of the third kind we have … ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

…

25: 1.8 Fourier Series

… ►Here

c_{n}

is related to

a_{n}

and

b_{n}

in (1.8.1), (1.8.2) by

c_{n}=\frac{1}{2}(a_{n}-\mathrm{i}b_{n})

c_{-n}=\frac{1}{2}(a_{n}+\mathrm{i}b_{n})

for

n>0

and

c_{0}=\frac{1}{2}a_{0}

. … ►As

n\to\infty

… ►(1.8.10) continues to apply if either

a

b

or both are infinite and/or

f(x)

has finitely many singularities in

(a,b)

, provided that the integral converges uniformly (§1.5(iv)) at

a, b

, and the singularities for all sufficiently large

\lambda

. … ►Then the series (1.8.1) converges to the sum … ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to

\sqrt{2\pi}\mathscr{F}\left(f\right)\left(-2\pi n\right)

. …

26: 14.19 Toroidal (or Ring) Functions

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates

(\eta,\theta,\phi)

, which are related to Cartesian coordinates

(x,y,z)

by … ►

z=\frac{c\sin\theta}{\cosh\eta-\cos\theta},

… ►

§14.19(iii) Integral Representations

… ►

14.19.4

P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+m+\frac{1}{2}% \right)(\sinh\xi)^{m}}{2^{m}\pi^{1/2}\Gamma\left(n-m+\frac{1}{2}\right)\Gamma% \left(m+\frac{1}{2}\right)}\*\int_{0}^{\pi}\frac{(\sin\phi)^{2m}}{(\cosh\xi+% \cos\phi\sinh\xi)^{n+m+(1/2)}}\,\mathrm{d}\phi,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable
A&S Ref:: 8.11.2
Permalink:: http://dlmf.nist.gov/14.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §14.19(iii), §14.19 and Ch.14

►

14.19.5

\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(n+% \frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}\right)\Gamma\left(n-m+\frac{1% }{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(mt\right)}{(\cosh\xi+\cosh t% \sinh\xi)^{n+(1/2)}}\,\mathrm{d}t,

m<n+\tfrac{1}{2}

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $m$ : nonnegative integer, $n$ : nonnegative integer and $\xi>0$ : variable
A&S Ref:: 8.11.4 (modified and corrected)
Permalink:: http://dlmf.nist.gov/14.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §14.19(iii), §14.19 and Ch.14

…

27: 22.20 Methods of Computation

… ►and the inverse sine has its principal value (§4.23(ii)). … ►Four iterations of (22.20.1) lead to

c_{4}=\Sci{6.5}{-12}

. … ►

§22.20(vi) Related Functions

… ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute

\operatorname{am}\left(x,k\right)

. … ►For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …

28: 36.13 Kelvin’s Ship-Wave Pattern

… ►In a reference frame where the ship is at rest we use polar coordinates

r

and

\phi

with

\phi=0

in the direction of the velocity of the water relative to the ship. Then with

g

denoting the acceleration due to gravity, the wave height is approximately given by …The integral is of the form of the real part of (36.12.1) with

y=\phi

u=\theta

g=1

k=\rho

, and … ►When

\rho>1

, that is, everywhere except close to the ship, the integrand oscillates rapidly. …The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency

\omega

as a function of wavevector

\mathbf{k}

: …

29: 22.16 Related Functions

§22.16 Related Functions

… ►

Relation to Elliptic Integrals

… ►

Relation to Theta Functions

… ►

Relation to the Elliptic Integral $E\left(\phi,k\right)$

… ►

Definition

…

30: 14.17 Integrals

§14.17 Integrals

►

§14.17(i) Indefinite Integrals

… ►

§14.17(ii) Barnes’ Integral

… ►Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013). ►

§14.17(iv) Definite Integrals of Products

…