relation to RC-function

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21: 31.7 Relations to Other Functions

§31.7 Relations to Other Functions

â–º

§31.7(i) Reductions to the Gauss Hypergeometric Function

… â–ºThey are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). … â–º

§31.7(ii) Relations to Lamé Functions

… â–ºSimilar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities

\zeta=K

K+i{K^{\prime}}

, and

i{K^{\prime}}

, where

K â¡

and

{K^{\prime}}

are related to

k

as in §19.2(ii).

22: 7.18 Repeated Integrals of the Complementary Error Function

… â–º

§7.18(iv) Relations to Other Functions

… â–º

Hermite Polynomials

… â–º

Confluent Hypergeometric Functions

… â–º

Parabolic Cylinder Functions

… â–º

Probability Functions

…

23: 7.5 Interrelations

§7.5 Interrelations

… â–º

7.5.5

e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z\right)=\mathrm{g}\left(z\right)+i% \mathrm{f}\left(z\right).

â“˜

Symbols:: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\mathcal{F}\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
Referenced by:: §7.13(iv), §7.5, §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §7.5 and Ch.7

… â–º

7.5.7

\zeta=\tfrac{1}{2}\sqrt{\pi}(1\mp i)z,

â“˜

Defines:: $\zeta$ : change of variable (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
Referenced by:: §7.12(ii), §7.5
Permalink:: http://dlmf.nist.gov/7.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §7.5 and Ch.7

… â–º … …

24: 15.17 Mathematical Applications

… â–ºThis topic is treated in §§15.10 and 15.11. … â–ºFor harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). … â–ºQuadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … â–ºThe three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …

25: 16.18 Special Cases

§16.18 Special Cases

… â–ºThis is a consequence of the following relations: …

26: 6.5 Further Interrelations

§6.5 Further Interrelations

… â–º

6.5.1

E_{1}\left(-x\pm i0\right)=-\operatorname{Ei}\left(x\right)\mp i\pi,

â“˜

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit and $x$ : real variable
A&S Ref:: 5.1.7
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §6.5 and Ch.6

… â–º

6.5.4

\tfrac{1}{2}(\operatorname{Ei}\left(x\right)-E_{1}\left(x\right))=% \operatorname{Chi}\left(x\right)=\operatorname{Ci}\left(ix\right)-\tfrac{1}{2}% \pi i.

â“˜

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\operatorname{Chi}\left(\NVar{z}\right)$ : hyperbolic cosine integral, $\mathrm{i}$ : imaginary unit and $x$ : real variable
A&S Ref:: 5.2.24 (in modified form)
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §6.5 and Ch.6

… â–º

6.5.5

\operatorname{Si}\left(z\right)=\tfrac{1}{2}i(E_{1}\left(-iz\right)-E_{1}\left% (iz\right))+\tfrac{1}{2}\pi,

â“˜

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable
A&S Ref:: 5.2.21
Referenced by:: §6.12(ii), §6.5
Permalink:: http://dlmf.nist.gov/6.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §6.5 and Ch.6

…

27: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

â–º

§15.9(i) Orthogonal Polynomials

… â–º

Jacobi

… â–º

Legendre

… â–º

Meixner

…

28: George E. Andrews

… â–ºAndrews was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences (USA) in 2003. …Andrews served as President of the AMS from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012. … â–º

17 q-Hypergeometric and Related Functions

…

29: Peter Paule

… â–º 1958 in Ried im Innkreis, Austria) is Professor of Mathematics (successor to Bruno Buchberger), Director of the Research Institute for Symbolic Computation (RISC), and Director of the Doctoral Program on Computational Mathematics at the Johannes Kepler University, Linz, Austria. â–ºPaule’s main research interests are computer algebra and algorithmic mathematics, together with connections to combinatorics, special functions, number theory, and other related fields. … â–ºPaule was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …