DLMF: Untitled Document

… ►Other reductions of

\mathit{H\!\ell}

to a

{{}_{2}F_{1}}

, with at least one free parameter, exist iff the pair

(a,p)

takes one of a finite number of values, where

q=\alpha\beta p

. … ►

31.7.4

\mathit{H\!\ell}\left(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2},\alpha\beta(\tfrac{1}{% 2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}(\alpha+\beta+1),\tfrac{1}{3% }(\alpha+\beta+1);z\right)={{}_{2}F_{1}}\left(\tfrac{1}{3}\alpha,\tfrac{1}{3}% \beta;\tfrac{1}{3}(\alpha+\beta+1);1-\left(1-\left(\tfrac{3}{2}-i\tfrac{\sqrt{% 3}}{2}\right)z\right)^{3}\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, $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.7(i), §31.7 and Ch.31

… ►

q=-\tfrac{1}{4}ah,

►

\alpha=-\tfrac{1}{2}\nu,

►

\beta=\tfrac{1}{2}(\nu+1),

…

… ►

Maxima (free interactive system)

ⓘ

Notes:: System for the manipulation of symbolic and numerical expressions. Includes a collection of special functions. Utilizes exact fractions, arbitrary precision integers, and arbitrary precision floating point numbers.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

L. Moser and M. Wyman (1958b) Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

ⓘ

External Links:: Document, MathReview (A. Sade), ZentralBlatt
Referenced by:: §26.1, §26.8(vii), Notations S.
Encodings:: BibTeX

►

S. L. B. Moshier (1989) Methods and Programs for Mathematical Functions. Ellis Horwood Ltd., Chichester.

ⓘ

Notes:: Includes diskette
External Links:: ISBN 0-13-578998-2, MathReview, ZentralBlatt
Referenced by:: 2nd item, CEPHES (free C library).
Encodings:: BibTeX

… ►

MPFR (free C library)

ⓘ

Notes:: MPFR is a C library for multiple-precision floating-point computations with correct rounding. The main goal of MPFR is to provide a library for multiple-precision floating-point computation which is both efficient and has well-defined semantics. MPFR includes most of the elementary functions. It also implements the gamma function, and more special functions are planned in the future.
External Links:: MPFR
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

►

mpmath (free python library)

ⓘ

Notes:: Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.
External Links:: mpmath
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

… ►

H. Rademacher (1973) Topics in Analytic Number Theory. Springer-Verlag, New York.

ⓘ

Notes:: Edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169
External Links:: MathReview (H. Halberstam), ZentralBlatt
Referenced by:: §1.8(iv), §20.7(viii), §20.7(viii), Ch.24.
Encodings:: BibTeX

… ►

S. Ramanujan (1927) Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

ⓘ

Referenced by:: §24.7(i).
Encodings:: BibTeX

… ►

REDUCE (free interactive system)

ⓘ

Notes:: Symbolic and extended-precision general purpose computer algebra system.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

M. Reed and B. Simon (1980) Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis. Elsevier, New York.

ⓘ

Notes:: revised and expanded Edition from 1972,
External Links:: ISBN 10: 0-12-585050-6(vol. 1), MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

… ►

K. H. Rosen (2004) Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.

ⓘ

External Links:: ISBN 0-201-87073-8, MathReview (Norman J. Richert)
Referenced by:: §27.17.
Encodings:: BibTeX

…

… ►

E. Pairman (1919) Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

ⓘ

External Links:: ZentralBlatt
Referenced by:: §5.1, Notations F.
Encodings:: BibTeX

… ►

PARI-GP (free interactive system and C library)

ⓘ

Notes:: Computer algebra system designed for computations in number theory. Many transcendental functions are also included. PARI is also available as a C library to allow for faster computations. Originally developed by Henri Cohen and co-workers at the Université Bordeaux I.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

A. M. Parkhurst and A. T. James (1974) Zonal Polynomials of Order

1

Through

12

. In Selected Tables in Mathematical Statistics, H. L. Harter and D. B. Owen (Eds.), Vol. 2, pp. 199–388.

ⓘ

External Links:: MathReview
Referenced by:: §35.11.
Encodings:: BibTeX

… ►

S. Porubský (1998) Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

ⓘ

External Links:: ZentralBlatt
Referenced by:: §24.10(iii).
Encodings:: BibTeX

… ►

W. H. Press and S. A. Teukolsky (1990) Elliptic integrals. Computers in Physics 4 (1), pp. 92–96.

ⓘ

Referenced by:: 2nd item.
Encodings:: BibTeX

…

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for

F\left(\phi,k\right)

and

E\left(\phi,k\right)

, together with special cases. …

… ►This equation has regular singularities at

0,1,a,\infty

, with corresponding exponents

\{0,1-\gamma\}

,

\{0,1-\delta\}

,

\{0,1-\epsilon\}

,

\{\alpha,\beta\}

, respectively (§2.7(i)). … ►The total number of free parameters is six. … ►

w(z)=z^{1-\gamma}w_{1}(z)

satisfies (31.2.1) if

w_{1}

is a solution of (31.2.1) with transformed parameters

q_{1}=q+(a\delta+\epsilon)(1-\gamma)

;

\alpha_{1}=\alpha+1-\gamma

,

\beta_{1}=\beta+1-\gamma

,

\gamma_{1}=2-\gamma

. Next,

w(z)=(z-1)^{1-\delta}w_{2}(z)

satisfies (31.2.1) if

w_{2}

is a solution of (31.2.1) with transformed parameters

q_{2}=q+a\gamma(1-\delta)

;

\alpha_{2}=\alpha+1-\delta

,

\beta_{2}=\beta+1-\delta

,

\delta_{2}=2-\delta

. … ►For example,

w(z)=(1-z)^{-\alpha}\tilde{w}(z/(z-1))

, which arises from

\tilde{z}=z/(z-1)

, satisfies (31.2.1) if

\tilde{w}(\tilde{z})

is a solution of (31.2.1) with

z

replaced by

\tilde{z}

and transformed parameters

\tilde{a}=a/(a-1)

,

\tilde{q}=-(q-a\alpha\gamma)/(a-1)

;

\tilde{\beta}=\alpha+1-\delta

,

\tilde{\delta}=\alpha+1-\beta

. …

… ►This formulation gives the bounded and unbounded solutions from the same formula (22.19.3), for

k\geq 1

and

k\leq 1

, respectively. … ►

Case II: $V(x)=\frac{1}{2}x^{2}-\frac{1}{4}\beta x^{4}$

… ►

Case III: $V(x)=-\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}$

… ►with period

4K\left(k\right)/\sqrt{2\eta-1}

and modulus

k=1/\sqrt{2-\eta^{-1}}

. … ►The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …

… ►

L. Carlitz (1953) Some congruences for the Bernoulli numbers. Amer. J. Math. 75 (1), pp. 163–172.

ⓘ

External Links:: FullText, MathReview (A. L. Whiteman), ZentralBlatt
Referenced by:: §24.10(i).
Encodings:: BibTeX

… ►

CEPHES (free C library)

ⓘ

Notes:: This library treats about 180 different mathematical functions, including a substantial collection of probability integrals, Bessel functions, and higher transcendental functions. Implementations in single, double, and quad precisions are provided. See Moshier (1989).
External Links:: GAMS (package CEPHES)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

ⓘ

External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

… ►

CoStLy (free C-XSC library)

ⓘ

Notes:: Complex interval standard function library. For details see Neher (2007)
External Links:: CoStLy
Referenced by:: § ‣ Software Cross Index, M. Neher (2007).
Encodings:: BibTeX

… ►

Cunningham Project (website)

ⓘ

Notes:: Seeks to factor numbers of the form $b^{n}\pm 1$ for $b=2,3,5,6,7,10,11,12$ .
External Links:: Home Page
Referenced by:: 3rd item.
Encodings:: BibTeX

…

… ►

A. Adelberg (1996) Congruences of

p

-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

G. E. Andrews and D. Foata (1980) Congruences for the

q

-secant numbers. European J. Combin. 1 (4), pp. 283–287.

ⓘ

External Links:: ISSN 0195-6698, MathReview (J. E. Cigler), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

ⓘ

External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

… ►

T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

ⓘ

External Links:: MathReview (Giovanni Coppola), ZentralBlatt
Referenced by:: (25.16.2), (25.16.4), §25.16(i), §25.16.
Encodings:: BibTeX

… ►

Axiom (free interactive system) Center for Algorithms and Interactive Scientific Software.

ⓘ

Notes:: Symbolic and extended-precision general purpose computer algebra system.
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

… ►

K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

ⓘ

External Links:: ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

►

J. W. L. Glaisher (1940) Number-Divisor Tables. British Association Mathematical Tables, Vol. VIII, Cambridge University Press, Cambridge, England.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §27.21.
Encodings:: BibTeX

… ►

K. Goldberg, F. T. Leighton, M. Newman, and S. L. Zuckerman (1976) Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

ⓘ

External Links:: MathReview (M. S. Cheema), ZentralBlatt
Referenced by:: §26.21.
Encodings:: BibTeX

… ►

H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

ⓘ

External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

… ►

GSL (free C library) GNU Scientific Library The GNU Project.

ⓘ

Notes:: The GNU Scientific Library is a large general purpose numerical software library with broad coverage of elementary and special functions. Implementation is in double precision.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

Zelle Exchange Toll Free 1(205-892-1862) Contact Number

21—30 of 850 matching pages

21: 31.7 Relations to Other Functions

22: Bibliography M

23: Bibliography R

24: Bibliography P

25: 19.13 Integrals of Elliptic Integrals

26: 31.2 Differential Equations

27: 22.19 Physical Applications

Case II: $V(x)=\frac{1}{2}x^{2}-\frac{1}{4}\beta x^{4}$

Case III: $V(x)=-\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}$

28: Bibliography C

29: Bibliography

30: Bibliography G

Zelle Exchange Toll Free 1(205-892-1862) Contact Number

21—30 of 850 matching pages

21: 31.7 Relations to Other Functions

22: Bibliography M

23: Bibliography R

24: Bibliography P

25: 19.13 Integrals of Elliptic Integrals

26: 31.2 Differential Equations

27: 22.19 Physical Applications

Case II: V ⁡ ( x ) = 1 2 ⁢ x 2 − 1 4 ⁢ β ⁢ x 4

Case III: V ⁡ ( x ) = − 1 2 ⁢ x 2 + 1 4 ⁢ β ⁢ x 4

28: Bibliography C

29: Bibliography

30: Bibliography G

Case II: $V(x)=\frac{1}{2}x^{2}-\frac{1}{4}\beta x^{4}$

Case III: $V(x)=-\frac{1}{2}x^{2}+\frac{1}{4}\beta x^{4}$