relation to sine and cosine integrals

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11: 14.5 Special Values

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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). … ►

14.5.21 $${𝖯}_{-\frac{1}{2}}\left(\cos \theta \right)=\frac{2}{\pi }K\left(\sin \left(\frac{1}{2}\theta \right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable

A&S Ref:

8.13.9

Permalink:

http://dlmf.nist.gov/14.5.E21

Encodings:

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See also:

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

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§14.5(vi) Addendum to §14.5(ii): $\mu =0$, $\nu =2$

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12: 28.10 Integral Equations

§28.10 Integral Equations

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§28.10(i) Equations with Elementary Kernels

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§28.10(ii) Equations with Bessel-Function Kernels

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§28.10(iii) Further Equations

… ►For relations with variable boundaries see Volkmer (1983).

13: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

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See also:

Annotations for §19.6(ii), §19.6 and Ch.19

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14: Bibliography S

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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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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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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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Algorithm 911: multiple-precision exponential integral and related functions. ACM Trans. Math. Software 37 (4), pp. Art. 46, 16.

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External Links:

Document, ZentralBlatt

Referenced by:

5th item, §4.48(iii), 3rd item, §6.21(ii).

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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15: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Recursion relations for the $3$-$j$ symbols. Nuclear Physics A 113 (1), pp. 215–220.

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External Links:

Document

Referenced by:

§34.3(iii).

Encodings:

BibTeX

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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External Links:

ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

Encodings:

BibTeX

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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16: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(ii) Integral Representations

… ►The inner integral does not converge. … ►

Sine and Cosine Functions

… ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

17: 10.9 Integral Representations

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Poisson’s and Related Integrals

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Schläfli’s and Related Integrals

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Mehler–Sonine and Related Integrals

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§10.9(ii) Contour Integrals

… ►See Paris and Kaminski (2001, p. 116) for related results. …

18: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

§19.10(ii) Elementary Functions

… ►In each case when $y=1$, the quantity multiplying $R_C$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …

19: 19.7 Connection Formulas

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Legendre’s Relation

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§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

►There are three relations connecting $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\omega }^2,k)$, where ${\omega }^2$ is a rational function of ${\alpha }^2$. … ►The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ when ${\alpha }^2>{\csc }^2\varphi$ (see (19.6.5) for the complete case). … ►The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: …

20: 22.11 Fourier and Hyperbolic Series

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22.11.3 $$\mathrm{dn}(z,k)=\frac{\pi }{2K}+\frac{2\pi }{K}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.3

Permalink:

http://dlmf.nist.gov/22.11.E3

Encodings:

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See also:

Annotations for §22.11 and Ch.22

… ►Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and , ►

22.11.13 $${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

… ►A related hyperbolic series is …where $E^{\prime }=E^{\prime }\left(k\right)$ is defined by §19.2.9. …