relation%20to%20inverse%20Gudermannian%20function

(0.006 seconds)

21—30 of 1012 matching pages

21: 20.11 Generalizations and Analogs

… ►If both

m, n

are positive, then

G(m,n)

allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … ►However, in this case

q

is no longer regarded as an independent complex variable within the unit circle, because

k

is related to the variable

\tau=\tau(k)

of the theta functions via (20.9.2). … … ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

22: 7.18 Repeated Integrals of the Complementary Error Function

… ►

§7.18(iv) Relations to Other Functions

… ►

Hermite Polynomials

… ►

Confluent Hypergeometric Functions

… ►

Parabolic Cylinder Functions

… ►

Probability Functions

…

23: 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

…

24: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►

§26.4(i) Definitions

… ►It is also the number of

k

-dimensional lattice paths from

(0,0,\ldots,0)

(n_{1},n_{2},\ldots,n_{k})

. For

k=0,1

, the multinomial coefficient is defined to be

1

. … ►(The empty set is considered to have one permutation consisting of no cycles.) … ►

§26.4(iii) Recurrence Relation

…

25: 18.40 Methods of Computation

… ►See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain

w(x)

. … ►It is now necessary to take the limit

\varepsilon\to 0{+}

F(x^{\prime}+\mathrm{i}\varepsilon)

, and the imaginary part is the required Stieltjes–Perron inversion: …Results of low (

~{}2

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

20

. … ►Interpolation of the midpoints of the jumps followed by differentiation with respect to

x

yields a Stieltjes–Perron inversion to obtain

w^{\mathrm{RCP}}(x)

to a precision of

\sim 4

decimal digits for

N=120

. …

26: 8.26 Tables

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Pearson (1965) tabulates the function $I(u,p)$ ( $=P\left(p+1,u\right)$ ) for $p=-1(.05)0(.1)5(.2)50$ , $u=0(.1)u_{p}$ to 7D, where $I(u,u_{p})$ rounds off to 1 to 7D; also $I(u,p)$ for $p=-0.75(.01)-1$ , $u=0(.1)6$ to 5D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

27: 16.7 Relations to Other Functions

§16.7 Relations to Other Functions

… ►Further representations of special functions in terms of

{{}_{p}F_{q}}

functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of

{{}_{q+1}F_{q}}

functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).

28: 6.16 Mathematical Applications

… ►Hence, if

x

is fixed and

n\to\infty

, then

S_{n}(x)\to\frac{1}{4}\pi

0

, or

-\frac{1}{4}\pi

according as

0<x<\pi

x=0

, or

-\pi<x<0

; compare (6.2.14). … ►Hence if

x=\pi/(2n)

and

n\to\infty

, then the limiting value of

S_{n}(x)

overshoots

\frac{1}{4}\pi

by approximately 18%. … ► … ►If we assume Riemann’s hypothesis that all nonreal zeros of

\zeta\left(s\right)

have real part of

\tfrac{1}{2}

(§25.10(i)), then …where

\pi(x)

is the number of primes less than or equal to

x

. …

29: Bibliography

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

ⓘ

Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(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

… ►

F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

ⓘ

External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

…

30: Bibliography K

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

ⓘ

External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. Khare, A. Lakshminarayan, and U. Sukhatme (2003) Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt
Referenced by:: §22.9(iv), §22.9(v).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

ⓘ

External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

ⓘ

External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

…