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21: 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

…

22: 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

…

23: 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.

24: 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).

25: 9.9 Zeros

… ►On the real line,

\operatorname{Ai}\left(x\right)

\operatorname{Ai}'\left(x\right)

\operatorname{Bi}\left(x\right)

\operatorname{Bi}'\left(x\right)

each have an infinite number of zeros, all of which are negative. … ►

§9.9(ii) Relation to Modulus and Phase

… ►

§9.9(iii) Derivatives With Respect to $k$

… ►

§9.9(iv) Asymptotic Expansions

… ►

§9.9(v) Tables

…

26: 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

. …

27: 20.11 Generalizations and Analogs

… ►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 applications to rapidly convergent expansions for

\pi

see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). … ►A further development on the lines of Neville’s notation (§20.1) is as follows. … ►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. …

28: 12.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … ►These notations are due to Miller (1952, 1955). An older notation, due to Whittaker (1902), for

U\left(a,z\right)

D_{\nu}\left(z\right)

. The notations are related by

U\left(a,z\right)=D_{-a-\frac{1}{2}}\left(z\right)

. …

29: 6.11 Relations to Other Functions

§6.11 Relations to Other Functions

… ►

Incomplete Gamma Function

… ►

Confluent Hypergeometric Function

►

6.11.2

E_{1}\left(z\right)=e^{-z}U\left(1,1,z\right),

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral and $z$ : complex variable
Referenced by:: §6.11, 5th item, §6.6
Permalink:: http://dlmf.nist.gov/6.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.11, §6.11 and Ch.6

►

6.11.3

\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{i}$ : imaginary unit, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.11, 5th item
Permalink:: http://dlmf.nist.gov/6.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §6.11, §6.11 and Ch.6

30: Publications

DLMF Related Publications

… ►

B. V. Saunders and Q. Wang (1999) Using Numerical Grid Generation to Facilitate 3D Visualization of Complicated Mathematical Functions, Technical Report NISTIR 6413 (November 1999), National Institute of Standards and Technology.

… ►

B. V. Saunders and Q. Wang (2000) From 2D to 3D: Numerical Grid Generation and the Visualization of Complex Surfaces, Proceedings of the 7th International Conference on Numerical Grid Generation in Computational Field Simulations, Whistler, British Columbia, Canada, September 25-28, 2000.

… ►

B. V. Saunders and Q. Wang (2006) From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

… ►

B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.