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11: 3.8 Nonlinear Equations

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$f(x_0)f^{\prime \prime }(x_0)>0$ and $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign between $x_0$ and $\xi$ (monotonic convergence).

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, $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign in the interval $(x_0,x_1)$, and $\xi \in [x_0,x_1]$ (monotonic convergence after the first iteration).

… ►Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by … ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

12: 20.11 Generalizations and Analogs

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§20.11(iii) Ramanujan’s Change of Base

… ►These results are called Ramanujan’s changes of base. …

13: 36 Integrals with Coalescing Saddles

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14: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

15: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

16: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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17: 12.11 Zeros

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12.11.5 $$p_0(\zeta )=t(\zeta ),$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.6 $$p_1(\zeta )=\frac{t^3-6t}{24{(t^2-1)}^2}+\frac{5}{48{((t^2-1){\zeta }^3)}^{\frac{1}{2}}}.$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.8 $$q_0(\zeta )=t(\zeta ).$$

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Symbols:

$\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/12.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.9 $$u_{a,1}\sim 2^{\frac{1}{2}}\mu \left(1-\mathrm{1.85575\hspace{0.33em}708}{\mu }^{-4/3}-\mathrm{0.34438\hspace{0.33em}34}{\mu }^{-8/3}-\mathrm{0.16871\hspace{0.33em}5}{\mu }^{-4}-0.11414{\mu }^{-16/3}-0.0808{\mu }^{-20/3}-\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: real or complex parameter and $u_{a,s}$: zeros

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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18: 9.7 Asymptotic Expansions

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9.7.1 $$\zeta =\frac{2}{3}{z}^{3/2}.$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$z$: complex variable

Permalink:

http://dlmf.nist.gov/9.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(i), §9.7 and Ch.9

… ►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►

9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.20 $$R_n(z)={(-1)}^n\sum _{k=0}^{m-1}{(-1)}^k{u}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+R_{m,n}(z),$$

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Defines:

$R_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $u_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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9.7.21 $$S_n(z)={(-1)}^{n-1}\sum _{k=0}^{m-1}{(-1)}^k{v}_k\frac{G_{n-k}\left(2\zeta \right)}{{\zeta }^k}+S_{m,n}(z),$$

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Defines:

$S_n$: remainder function (locally)

Symbols:

$G_p\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index and $v_s$: expansion coefficient

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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19: Errata

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Changes

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Changes

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Changes

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Changes

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Changes

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20: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …