asymptotic approximations to zeros

SearchAdvancedHelp

(0.010 seconds)

21—30 of 66 matching pages

21: 8.22 Mathematical Applications

… ►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. … ►

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

… ►so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then …For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). ►The Debye functions ${\int }_0^x{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ and ${\int }_x^{\mathrm{\infty }}{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ are closely related to the incomplete Riemann zeta function and the Riemann zeta function. …

22: 18.32 OP’s with Respect to Freud Weights

§18.32 OP’s with Respect to Freud Weights

… ►where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime }(x)$ increases for $x>0$, and $Q^{\prime }(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$, see Freud (1969). …However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). …

23: 2.8 Differential Equations with a Parameter

… ►Zeros of $f(z)$ are also called turning points. … ►In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order $1/(\lambda +2)$. … ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. ►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). ►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv). …

24: 2.4 Contour Integrals

… ►

§2.4(i) Watson’s Lemma

… ►Furthermore, as $t\to 0+$, $q(t)$ has the expansion (2.3.7). … ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

… ►

§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

…

25: 25.16 Mathematical Applications

… ►In studying the distribution of primes $p\le x$, Chebyshev (1851) introduced a function $\psi \left(x\right)$ (not to be confused with the digamma function used elsewhere in this chapter), given by …which is related to the Riemann zeta function by …where the sum is taken over the nontrivial zeros $\rho$ of $\zeta \left(s\right)$. ►The prime number theorem (27.2.3) is equivalent to the statement … ►The Riemann hypothesis is equivalent to the statement …

26: Bibliography C

… ►

Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ 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

… ►

Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.12, §19.27(ii), §19.27(iii), §19.27(iv), §19.27(v), §19.27(vi).

Encodings:

BibTeX

… ►

Asymptotics of the largest zeros of some orthogonal polynomials. J. Phys. A 31 (25), pp. 5525–5544.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (Tim H. Baker), ZentralBlatt

Referenced by:

§18.26(v), §18.29.

Encodings:

BibTeX

►

Introduction to Approximation Theory. 2nd edition, Chelsea Publishing Co., New York.

ⓘ

Notes:

Reprinted by AMS Chelsea Publishing, Providence, RI, 1998

External Links:

ISBN 0-8218-1374-9, MathReview, ZentralBlatt

Referenced by:

§18.38(i).

Encodings:

BibTeX

… ►

Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions. IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.

ⓘ

External Links:

ISSN 0018-9480, FullText

Referenced by:

§10.21(x).

Encodings:

BibTeX

…

27: 10.72 Mathematical Applications

… ►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of $z_0$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). These approximations are uniform with respect to both $z$ and $\alpha$, including $z=z_0(a)$, the cut neighborhood of $z=0$, and $\alpha =a$. …

28: 12.11 Zeros

§12.11 Zeros

… ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

29: Bibliography B

… ►

The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

ⓘ

External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

BibTeX

… ►

The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41 (2), pp. 236–266.

ⓘ

External Links:

ISSN 1095-7200, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§25.17.

Encodings:

BibTeX

… ►

Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 (2), pp. 314–337.

ⓘ

External Links:

ISSN 0047-259X, Document, MathReview (A. L. Rukhin), ZentralBlatt

Referenced by:

§35.10, §35.9.

Encodings:

BibTeX

… ►

Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.

ⓘ

External Links:

ISSN 0022-2526, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§18.35(iii).

Encodings:

BibTeX

… ►

Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Dan Polis̆evski), ZentralBlatt

Referenced by:

§2.4(iv).

Encodings:

BibTeX

…

30: 18.35 Pollaczek Polynomials

… ►The type 2 polynomials reduce for $a=b=0$ to ultraspherical polynomials, see (18.35.8). … ►we have the explicit representations … ►For type 3 orthogonality (18.35.5) generalizes to … ►See Bo and Wong (1996) for an asymptotic expansion of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$ as $n\to \mathrm{\infty }$, with $a$ and $b$ fixed. …Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$. …