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21: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►For multiple zeros the convergence is linear, but if the multiplicity $m$ is known then quadratic convergence can be restored by multiplying the ratio $f(z_n)/f^{\prime }(z_n)$ in (3.8.4) by $m$. … ►has $n$ zeros in $ℂ$, counting each zero according to its multiplicity. … ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ► …

22: Bibliography C

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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External Links:

MathReview (K. Goldberg), ZentralBlatt

Referenced by:

§24.6.

Encodings:

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New proof of the addition theorem for Gegenbauer polynomials. SIAM J. Math. Anal. 2, pp. 347–351.

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External Links:

ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§18.18(ii).

Encodings:

BibTeX

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.26(i).

Encodings:

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The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.

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External Links:

ISSN 0002-9947, FullText, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Modular Forms and Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-94609-8; 0-387-98998-6, MathReview (M. Ram Murty), ZentralBlatt

Referenced by:

§23.20(v).

Encodings:

BibTeX

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23: 4.21 Identities

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§4.21(iii) Multiples of the Argument

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De Moivre’s Theorem

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24: 2.1 Definitions and Elementary Properties

… ►For example, if $f(z)$ is analytic for all sufficiently large $|z|$ in a sector $𝐒$ and $f(z)=O\left(z^{\nu }\right)$ as $z\to \mathrm{\infty }$ in $𝐒$, $\nu$ being real, then $f^{\prime }(z)=O\left(z^{\nu -1}\right)$ as $z\to \mathrm{\infty }$ in any closed sector properly interior to $𝐒$ and with the same vertex (Ritt’s theorem). This result also holds with both $O$’s replaced by $o$’s. … ►These include addition, subtraction, multiplication, and division. … ►Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …

25: Errata

… ►We now include Markov’s Theorem. In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. … ►

17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}}$$

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

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The multi-product notation ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

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17.2.22 $$\frac{{(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n}{{(a^{\frac{1}{2}},-a^{\frac{1}{2}};q)}_n}=\frac{{(a{q}^2;q^2)}_n}{{(a;q^2)}_n}=\frac{1-a{q}^{2n}}{1-a}$$

17.2.23 $$\frac{{(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}{{(a^{\frac{1}{k}},{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n}=\frac{{(a{q}^k;q^k)}_n}{{(a;q^k)}_n}=\frac{1-a{q}^{kn}}{1-a}$$

The numerators of the leftmost fractions were corrected to read ${(q{a}^{\frac{1}{2}},-q{a}^{\frac{1}{2}};q)}_n$ and ${(q{a}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$ instead of ${(q{a}^{\frac{1}{2}},-a{q}^{\frac{1}{2}};q)}_n$ and ${(a{q}^{\frac{1}{k}},q{\omega }_k{a}^{\frac{1}{k}},\mathrm{\dots },q{\omega }_k^{k-1}{a}^{\frac{1}{k}};q)}_n$, respectively.

Reported 2017-06-26 by Jason Zhao.

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26: 1.5 Calculus of Two or More Variables

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Implicit Function Theorem

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§1.5(iii) Taylor’s Theorem; Maxima and Minima

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§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

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§1.5(v) Multiple Integrals

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27: Bibliography K

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On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

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Notes:

In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(i), §10.74(vi).

Encodings:

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On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

External Links:

ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

Encodings:

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Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).

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Notes:

In Russian. English translation: Soviet Phys. Dokl. 33(3),196–198, (1988).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi).

Encodings:

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A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.10.

Encodings:

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

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28: 18.27 $q$-Hahn Class

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18.27.9 $$v_x=\frac{{(a^{-1}x,c^{-1}x;q)}_{\mathrm{\infty }}}{{(x,b{c}^{-1}x;q)}_{\mathrm{\infty }}},$$ , , ,

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_x$

Permalink:

http://dlmf.nist.gov/18.27.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.9_5 $$h_n=\frac{{\left(-c\right)}^n{a}^{n+1}}{1-ab{q}^{2n+1}}\frac{{(q;q)}_n{q}^{\left(\genfrac{}{}{0.0pt}{}{n+2}{2}\right)}}{{(aq,cq;q)}_n}\frac{{(q,c^{-1}aq,a^{-1}c,ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq,cq,b{q}^{n+1},c^{-1}ab{q}^{n+1};q)}_{\mathrm{\infty }}},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$

Referenced by:

§18.27(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E9_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.12 $$v_x=\frac{{(qx/c,-qx/d;q)}_{\mathrm{\infty }}}{{(q^{\alpha +1}x/c,-q^{\beta +1}x/d;q)}_{\mathrm{\infty }}},$$ $\alpha ,\beta >-1$, $c,d>0$.

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $x$: real variable and $v_x$

Permalink:

http://dlmf.nist.gov/18.27.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

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18.27.14_1 $$h_n=\frac{{\left(aq\right)}^n}{1-ab{q}^{2n+1}}\frac{{(q,bq;q)}_n}{{(aq;q)}_n}\frac{{(ab{q}^{n+1};q)}_{\mathrm{\infty }}}{{(aq;q)}_{\mathrm{\infty }}}.$$

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $n$: nonnegative integer and $h_n$

Source:

Koekoek et al. (2010, (14.12.2))

Referenced by:

§18.27(iv), (18.28.27), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E14_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(iv), §18.27 and Ch.18

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18.27.17_2 $$h_0^{(2)}=\frac{{(q,-c{q}^{\alpha +1},-c^{-1}{q}^{-\alpha };q)}_{\mathrm{\infty }}}{{(q^{\alpha +1},-c,-c^{-1}q;q)}_{\mathrm{\infty }}}.$$

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Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable and $h_n$

Source:

Koekoek et al. (2010, (14.21.3))

Permalink:

http://dlmf.nist.gov/18.27.E17_2

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(v), §18.27 and Ch.18

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29: Bibliography M

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Asymptotic expansion of a multiple integral. SIAM J. Math. Anal. 18 (6), pp. 1630–1637.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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External Links:

ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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External Links:

ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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External Links:

ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

Encodings:

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On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.

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External Links:

Document, MathReview (V. F. Cowling), ZentralBlatt

Referenced by:

(25.6.5).

Encodings:

BibTeX

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