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1: 5.4 Special Values and Extrema

§5.4 Special Values and Extrema

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§5.4(iii) Extrema

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5.4.20 $$x_n=-n+\frac{1}{\pi }\arctan \left(\frac{\pi }{\ln n}\right)+O\left(\frac{1}{n{(\ln n)}^2}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

6.3.20 (The error estimate has been improved.)

Referenced by:

§5.4(iii)

Permalink:

http://dlmf.nist.gov/5.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(iii), §5.4 and Ch.5

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2: 18.4 Graphics

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Figure 18.4.2: Jacobi polynomials $P_n^{(1.25,0.75)}\left(x\right)$, $n=7,8$. This illustrates inequalities for extrema of a Jacobi polynomial; see (18.14.16). … Magnify

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3: Bibliography W

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Asymptotic monotonicity of the relative extrema of Jacobi polynomials. Canad. J. Math. 46 (6), pp. 1318–1337.

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External Links:

ISSN 0008-414X, MathReview (S. I. Drobnies), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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On the relative extrema of the Jacobi polynomials $P_n^{(0,-1)}(x)$ . SIAM J. Math. Anal. 25 (2), pp. 776–811.

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External Links:

ISSN 0036-1410, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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4: Bibliography S

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On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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On the relative extrema of the Hermite orthogonal functions. J. Indian Math. Soc. (N.S.) 15, pp. 129–134.

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External Links:

MathReview (J. Todd), ZentralBlatt

Referenced by:

§18.14(i).

Encodings:

BibTeX

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5: 3.5 Quadrature

… ►For the latter $a=-1$, $b=1$, and the nodes $x_k$ are the extrema of the Chebyshev polynomial $T_n\left(x\right)$ (§3.11(ii) and §18.3). …

6: Errata

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The table of extrema for the Euler gamma function $\mathrm{\Gamma }$ had several entries in the $x_n$ column that were wrong in the last 2 or 3 digits. These have been corrected and 10 extra decimal places have been included.

$n$	$x_n$	$\mathrm{\Gamma }\left(x_n\right)$
0	$\mathrm{1.46163\hspace{0.33em}21449\hspace{0.33em}68362\hspace{0.33em}34126}$	$\mathrm{0.88560\hspace{0.33em}31944\hspace{0.33em}10888\hspace{0.33em}70028}$
1	$-\mathrm{0.50408\hspace{0.33em}30082\hspace{0.33em}64455\hspace{0.33em}40926}$	$-\mathrm{3.54464\hspace{0.33em}36111\hspace{0.33em}55005\hspace{0.33em}08912}$
2	$-\mathrm{1.57349\hspace{0.33em}84731\hspace{0.33em}62390\hspace{0.33em}45878}$	$\mathrm{2.30240\hspace{0.33em}72583\hspace{0.33em}39680\hspace{0.33em}13582}$
3	$-\mathrm{2.61072\hspace{0.33em}08684\hspace{0.33em}44144\hspace{0.33em}65000}$	$-\mathrm{0.88813\hspace{0.33em}63584\hspace{0.33em}01241\hspace{0.33em}92010}$
4	$-\mathrm{3.63529\hspace{0.33em}33664\hspace{0.33em}36901\hspace{0.33em}09784}$	$\mathrm{0.24512\hspace{0.33em}75398\hspace{0.33em}34366\hspace{0.33em}25044}$
5	$-\mathrm{4.65323\hspace{0.33em}77617\hspace{0.33em}43142\hspace{0.33em}44171}$	$-\mathrm{0.05277\hspace{0.33em}96395\hspace{0.33em}87319\hspace{0.33em}40076}$
6	$-\mathrm{5.66716\hspace{0.33em}24415\hspace{0.33em}56885\hspace{0.33em}53585}$	$\mathrm{0.00932\hspace{0.33em}45944\hspace{0.33em}82614\hspace{0.33em}85052}$
7	$-\mathrm{6.67841\hspace{0.33em}82130\hspace{0.33em}73426\hspace{0.33em}74283}$	$-\mathrm{0.00139\hspace{0.33em}73966\hspace{0.33em}08949\hspace{0.33em}76730}$
8	$-\mathrm{7.68778\hspace{0.33em}83250\hspace{0.33em}31626\hspace{0.33em}03744}$	$\mathrm{0.00018\hspace{0.33em}18784\hspace{0.33em}44909\hspace{0.33em}40419}$
9	$-\mathrm{8.69576\hspace{0.33em}41638\hspace{0.33em}16401\hspace{0.33em}26649}$	$-\mathrm{0.00002\hspace{0.33em}09252\hspace{0.33em}90446\hspace{0.33em}52667}$
10	$-\mathrm{9.70267\hspace{0.33em}25400\hspace{0.33em}01863\hspace{0.33em}73608}$	$\mathrm{0.00000\hspace{0.33em}21574\hspace{0.33em}16104\hspace{0.33em}52285}$

Reported 2018-02-17 by David Smith.

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