generalized sine and cosine integrals

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11: 22.14 Integrals

§22.14 Integrals

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§22.14(i) Indefinite Integrals of Jacobian Elliptic Functions

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§22.14(iii) Other Indefinite Integrals

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§22.14(iv) Definite Integrals

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12: 15.17 Mathematical Applications

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§15.17(ii) Conformal Mappings

… ►Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. … ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …

13: 14.5 Special Values

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14.5.11 $${𝖯}_{\nu }^{1/2}\left(\cos \theta \right)={\left(\frac{2}{\pi \sin \theta }\right)}^{1/2}\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.12

Permalink:

http://dlmf.nist.gov/14.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

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14.5.14 $${𝖰}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\frac{\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}.$$

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Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.15 (corrected)

Referenced by:

Erratum (V1.0.16) for Equation (14.5.14)

Permalink:

http://dlmf.nist.gov/14.5.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

Originally this equation was incorrect because of a minus sign in front of the right-hand side.

Reported 2017-04-10 by André Greiner-Petter

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

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§14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

►In this subsection $K\left(k\right)$ and $E\left(k\right)$ denote the complete elliptic integrals of the first and second kinds; see §19.2(ii). …

14: 4.37 Inverse Hyperbolic Functions

… ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …The principal branches are denoted by $\mathrm{arcsinh}$, $\mathrm{arccosh}$, $\mathrm{arctanh}$ respectively. Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

Inverse Hyperbolic Sine

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Inverse Hyperbolic Cosine

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15: 14.12 Integral Representations

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14.12.1 $${𝖯}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{2^{1/2}{(\sin \theta )}^{\mu }}{{\pi }^{1/2}\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{\int }_0^{\theta }\frac{\cos \left(\left(\nu +\frac{1}{2}\right)t\right)}{{(\cos t-\cos \theta )}^{\mu +(1/2)}}dt,$$ , .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.20(iv), §18.10(i)

Permalink:

http://dlmf.nist.gov/14.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

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14.12.3 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){(\sin \theta )}^{\mu }}{2^{\mu +1}\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}\left({\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta +\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt+{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{(\cos \theta -\mathrm{i}\sin \theta \cosh t)}^{\nu +\mu +1}}dt\right),$$ , $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }\nu \pm \mu >-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(i), §14.12(i), §14.12 and Ch.14

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14.12.6 $${𝑸}_{\nu }^{\mu }\left(x\right)=\frac{{\pi }^{1/2}{\left(x^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(x+{(x^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.8.2 (modified)

Permalink:

http://dlmf.nist.gov/14.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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16: 4.23 Inverse Trigonometric Functions

… ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. …The principal branches are denoted by $\arcsin z$, $\arccos z$, $\arctan z$, respectively. Each is two-valued on the corresponding cuts, and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts. … ►

Inverse Sine

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Inverse Cosine

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17: Bibliography B

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Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.

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External Links:

Document, ZentralBlatt

Referenced by:

§19.2(iii).

Encodings:

BibTeX

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Integral transforms with generalized Legendre functions as kernels. Compositio Math. 18, pp. 235–287.

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External Links:

MathReview (R. K. Saxena), ZentralBlatt

Referenced by:

§14.20(vi), §14.29, §14.31(ii).

Encodings:

BibTeX

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Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.

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External Links:

ISBN 978-0-12-373624-6; 0-12-373624-2, MathReview (Gerd Teschke)

Referenced by:

§18.3.

Encodings:

BibTeX

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

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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

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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

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18: Bibliography S

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A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals. Comput. Phys. Comm. 66 (2-3), pp. 271–275.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

Encodings:

BibTeX

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Numerical evaluation of integrals of the form $I={\int }_{x_1}^{x_2}f(x)e^{i\varphi (x)}𝑑x$ and the tabulation of the function $\mathrm{Gi}(z)=(1/\pi ){\int }_0^{\mathrm{\infty }}\sin (uz+\frac{1}{3}{u}^3)𝑑u$ . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Complements to asymptotic development of sine cosine integrals, and auxiliary functions. Univ. Beograd. Publ. Elecktrotehn. Fak., Ser. Mat. Fiz. 461–497, pp. 185–191.

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External Links:

MathReview (E. Riekstins), ZentralBlatt

Referenced by:

§6.15, §6.15.

Encodings:

BibTeX

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Automatic computing methods for special functions. III. The sine, cosine, exponential integrals, and related functions. J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 291–311.

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Notes:

Includes Fortran code, maximum accuracy 18S.

External Links:

MathReview (Martin E. Price), ZentralBlatt

Referenced by:

§6.21(ii).

Encodings:

BibTeX

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19: 19.23 Integral Representations

§19.23 Integral Representations

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19.23.6 $$4\pi R_F(x,y,z)={\int }_0^{2\pi }{\int }_0^{\pi }\frac{\sin \theta d\theta d\varphi }{{(x{\sin }^2\theta {\cos }^2\varphi +y{\sin }^2\theta {\sin }^2\varphi +z{\cos }^2\theta )}^{1/2}},$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.16(ii), §19.23

Permalink:

http://dlmf.nist.gov/19.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

… ►In (19.23.8)–(19.23.10) one or more of the variables may be 0 if the integral converges. … ►With $l_1,l_2,l_3$ denoting any permutation of $\sin \theta \cos \varphi$, $\sin \theta \sin \varphi$, $\cos \theta$, … ►For generalizations of (19.23.6_5) and (19.23.8) see Carlson (1964, (6.2), (6.12), and (6.1)).

20: 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 $\cos \varphi$ are assumed to be nonnegative. …More generally in (19.14.4), … ►

§19.14(ii) General Case

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