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11—20 of 215 matching pages

11: 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. …

12: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

…

13: Tom M. Apostol

… ►Apostol was born on August 20, 1923. He received his bachelor of science in chemical engineering in 1944 and a master’s degree in mathematics in 1946, both from the University of Washington, Seattle. …

14: 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

\pmod{m}

, which is correct to 20 digits. …

15: 6.19 Tables

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Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

16: Peter L. Walker

… ►

20 Theta Functions

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17: Staff

… ►

William P. Reinhardt, University of Washington, Chaps. 20, 22, 23

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Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23

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William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23

… ►

Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23

…

18: 28.35 Tables

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Ince (1932) includes eigenvalues $a_{n}$ , $b_{n}$ , and Fourier coefficients for $n=0$ or $1(1)6$ , $q=0(1)10(2)20(4)40$ ; 7D. Also $\operatorname{ce}_{n}\left(x,q\right)$ , $\operatorname{se}_{n}\left(x,q\right)$ for $q=0(1)10$ , $x=1(1)90$ , corresponding to the eigenvalues in the tables; 5D. Notation: $a_{n}=\mathit{be}_{n}-2q$ , $b_{n}=\mathit{bo}_{n}-2q$ .

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Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$ , $\operatorname{Se}_{n+1}\left(x,q\right)$ for $n=0(1)5$ , $q=1(1)20$ , $x=0.1(.1)1$ ; 4D or 5D.

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•

National Bureau of Standards (1967) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n}\left(q\right)$ for $n=0(1)3$ with $q=0(.2)20(.5)37(1)100$ , and $n=4(1)15$ with $q=0(2)100$ ; Fourier coefficients for $\operatorname{ce}_{n}\left(x,q\right)$ and $\operatorname{se}_{n}\left(x,q\right)$ for $n=0(1)15$ , $n=1(1)15$ , respectively, and various values of $q$ in the interval $[0,100]$ ; joining factors $g_{\mathit{e},n}(\sqrt{q})$ , $f_{\mathit{e},n}(\sqrt{q})$ for $n=0(1)15$ with $q=0(.5\mbox{ to }10)100$ (but in a different notation). Also, eigenvalues for large values of $q$ . Precision is generally 8D.

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Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$ , $b_{n+1}\left(q\right)$ for $n=0(1)4$ , $q=0(1)50$ ; $n=0(1)20$ ( $a$ ’s) or 19 ( $b$ ’s), $q=1,3,5,10,15,25,50(50)200$ . Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , $n=0(1)7$ . Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n+1}\left(x,10\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $x=0(5^{\circ})90^{\circ}$ . Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$ , and their first $x$ -derivatives for $n=0(1)4$ , $j=1,2$ , $x=0(.2)4$ . Precision is mostly 9S.

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Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of $\operatorname{ce}_{n}\left(x,10\right)$ , $\operatorname{se}_{n}\left(x,10\right)$ for $n=1(1)10$ , and the first 5 zeros of ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ , ${\operatorname{Ms}^{(j)}_{n}}\left(x,\sqrt{10}\right)$ for $n=0$ or $1(1)8$ , $j=1,2$ . Precision is mostly 9S.

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19: 10.75 Tables

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•

Bickley et al. (1952) tabulates $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ or $x^{n}Y_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ ( $.01$ or $.1$ ) $10(.1)25$ , 8D (for $J_{n}\left(x\right)$ ), 8S (for $Y_{n}\left(x\right)$ or $x^{n}Y_{n}\left(x\right)$ ); $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)25$ , 10D (for $J_{n}\left(x\right)$ ), 10S (for $Y_{n}\left(x\right)$ ).

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The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_{0}\left(x\right)$ to 15D, $J_{1}\left(x\right)$ , $J_{2}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ to 10D, $Y_{2}\left(x\right)$ to 8D, $x=0(.1)17.5$ ; $Y_{n}\left(x\right)-(2/\pi)J_{n}\left(x\right)\ln x$ , $n=0,1$ , $x=0(.1)2$ , 8D; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)20$ , 5D or 5S; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 10S; modulus and phase functions $\sqrt{x}M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)-x$ , $n=0,1,2$ , $\ifrac{1}{x}=0(.01)0.1$ , 8D.

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Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

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Zhang and Jin (1996, pp. 185–195) tabulates $J_{n}\left(x\right)$ , $J_{n}'\left(x\right)$ , $Y_{n}\left(x\right)$ , $Y_{n}'\left(x\right)$ , $n=0(1)10(10)50,100$ , $x=1$ , 5, 10, 25, 50, 100, 9S; $J_{n+\alpha}\left(x\right)$ , $J_{n+\alpha}'\left(x\right)$ , $Y_{n+\alpha}\left(x\right)$ , $Y_{n+\alpha}'\left(x\right)$ , $n=0(1)5,10,30,50,100$ , $\alpha=\tfrac{1}{4},\tfrac{1}{3},\tfrac{1}{2},\tfrac{2}{3},\tfrac{3}{4}$ , $x=1,5,10,50$ , 8S; real and imaginary parts of $J_{n+\alpha}\left(z\right)$ , $J_{n+\alpha}'\left(z\right)$ , $Y_{n+\alpha}\left(z\right)$ , $Y_{n+\alpha}'\left(z\right)$ , $n=0(1)15,20(10)50,100$ , $\alpha=0,\tfrac{1}{2}$ , $z=4+2i$ , $20+10i$ , 8S.

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Olver (1960) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(\tfrac{1}{2})20\tfrac{1}{2}$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ ; see §10.21(viii), and more fully Olver (1954).

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20: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …