relation to zeros

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11: 8.22 Mathematical Applications

… ►

§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

… ►so that

\lim_{x\to\infty}\zeta_{x}(s)=\zeta\left(s\right)

, then ►

8.22.3

\zeta_{x}(s)=\sum_{k=1}^{\infty}k^{-s}P\left(s,kx\right),

\Re s>1

ⓘ

Symbols:: $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $\Re$ : real part, $x$ : real variable, $k$ : nonnegative integer and $\zeta_{x}(s)$ : incomplete Riemann zeta function
Permalink:: http://dlmf.nist.gov/8.22.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §8.22(ii), §8.22 and Ch.8

►For further information on

\zeta_{x}(s)

, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). ►The Debye functions

\int_{0}^{x}t^{n}\left(e^{t}-1\right)^{-1}\,\mathrm{d}t

and

\int_{x}^{\infty}t^{n}\left(e^{t}-1\right)^{-1}\,\mathrm{d}t

are closely related to the incomplete Riemann zeta function and the Riemann zeta function. …

12: 13.9 Zeros

§13.9 Zeros

… ► … ► … ► … ►For fixed

a

and

z

\mathbb{C}

U\left(a,b,z\right)

has two infinite strings of

b

-zeros that are asymptotic to the imaginary axis as

|b|\to\infty

… ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ► … ►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … ►are related to Bessel polynomials (§§10.49(ii) and 18.34). … …

14: Bibliography C

… ►

F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial

{L}_{n}^{\alpha}(x)

as the index

\alpha\rightarrow\infty

and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

ⓘ

External Links:: ISSN 0024-1318, Document, MathReview (R. A. Askey)
Referenced by:: §18.7(iii).
Encodings:: BibTeX

…

15: Bibliography I

… ►

Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of

J_{0}(z)-iJ_{1}(z)

and of Bessel functions

J_{m}(z)

of any real order

m

. Linear Algebra Appl. 194, pp. 35–70.

ⓘ

External Links:: ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

►

Y. Ikebe, Y. Kikuchi, and I. Fujishiro (1991) Computing zeros and orders of Bessel functions. J. Comput. Appl. Math. 38 (1-3), pp. 169–184.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vi), §3.8(iii).
Encodings:: BibTeX

… ►

M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions,

q

-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

ⓘ

External Links:: ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

►

M. E. H. Ismail and D. R. Masson (1991) Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

ⓘ

External Links:: ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt
Referenced by:: §18.30(i).
Encodings:: BibTeX

►

M. E. H. Ismail and M. E. Muldoon (1995) Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

ⓘ

External Links:: ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.21(v).
Encodings:: BibTeX

…

16: 18.27 $q$ -Hahn Class

… ►Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … ►

From Big $q$ -Jacobi to Jacobi

… ►

From Big $q$ -Jacobi to Little $q$ -Jacobi

… ►

From Little $q$ -Jacobi to Jacobi

… ►

From Little $q$ -Laguerre to Laguerre

…

17: 22.2 Definitions

… ►

22.2.3

\zeta=\frac{\pi z}{2K\left(k\right)},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►

22.2.9

\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.

ⓘ

Defines:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►Each is meromorphic in

z

for fixed

k

, with simple poles and simple zeros, and each is meromorphic in

k

for fixed

z

. … ►The Jacobian functions are related in the following way. … ►and on the left-hand side of (22.2.11)

\mathrm{p}

\mathrm{q}

are any pair of the letters

\mathrm{s}

\mathrm{c}

\mathrm{d}

\mathrm{n}

, and on the right-hand side they correspond to the integers

1,2,3,4

18: 1.10 Functions of a Complex Variable

… ►

Zeros

… ► … ►each location again being counted with multiplicity equal to that of the corresponding zero or pole. … ►(The integer

k

may be greater than one to allow for a finite number of zero factors.) … ►

§1.10(x) Infinite Partial Fractions

…

19: 25.18 Methods of Computation

… ►

§25.18(i) Function Values and Derivatives

… ►Calculations relating to derivatives of

\zeta\left(s\right)

and/or

\zeta\left(s,a\right)

can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). … ►

§25.18(ii) Zeros

►Most numerical calculations of the Riemann zeta function are concerned with locating zeros of

\zeta\left(\frac{1}{2}+it\right)

in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of

\zeta\left(s\right)

lie on the critical line

\Re s=\frac{1}{2}

. Calculations to date (2008) have found no nontrivial zeros off the critical line. …

20: 24.12 Zeros

§24.12 Zeros

►

§24.12(i) Bernoulli Polynomials: Real Zeros

… ►

§24.12(ii) Euler Polynomials: Real Zeros

… ►

§24.12(iii) Complex Zeros

… ►

§24.12(iv) Multiple Zeros

…