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B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

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Q. Wang and B. V. Saunders (2005) Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

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A. Youssef (2007) Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007.

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B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

3: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

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4: 13.30 Tables

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Slater (1960) tabulates $M\left(a,b,x\right)$ for $a=-1(.1)1$ , $b=0.1(.1)1$ , and $x=0.1(.1)10$ , 7–9S; $M\left(a,b,1\right)$ for $a=-11(.2)2$ and $b=-4(.2)1$ , 7D; the smallest positive $x$ -zero of $M\left(a,b,x\right)$ for $a=-4(.1){-}0.1$ and $b=0.1(.1)2.5$ , 7D.

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Zhang and Jin (1996, pp. 411–423) tabulates $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ for $a=-5(.5)5$ , $b=0.5(.5)5$ , and $x=0.1,1,5,10,20,30$ , 8S (for $M\left(a,b,x\right)$ ) and 7S (for $U\left(a,b,x\right)$ ).

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5: 24.2 Definitions and Generating Functions

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Table 24.2.3: Bernoulli numbers

B_{n}=N/D

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$n$	$N$	$D$	$B_{n}$
…
$30$	$861\;58412\;76005$	$14322$	$6.01580\;8739$	$\times 10^{8}$
…

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Table 24.2.4: Euler numbers

E_{n}

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$n$	$E_{n}$
…
$30$	$-44\;15438\;93249\;02310\;45536\;82821$
…

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Table 24.2.5: Coefficients

b_{n,k}

of the Bernoulli polynomials

B_{n}\left(x\right)=\sum_{k=0}^{n}b_{n,k}x^{k}

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…
$11$	$0$	$\tfrac{5}{6}$	$0$	$-\tfrac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\tfrac{55}{6}$	$-\tfrac{11}{2}$	$1$
…

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Table 24.2.6: Coefficients

e_{n,k}

of the Euler polynomials

E_{n}\left(x\right)=\sum_{k=0}^{n}e_{n,k}x^{k}

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	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
…

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6: 26.2 Basic Definitions

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Table 26.2.1: Partitions

p\left(n\right)

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
6	11	23	1255	40	37338
…
9	30	26	2436	43	63261
…
11	56	28	3718	45	89134
…
13	101	30	5604	47	1 24754
…

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7: Staff

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Richard B. Paris, University of Abertay, Chaps. 8, 11

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Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

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Richard B. Paris, University of Abertay Dundee, for Chaps. 8, 11 (deceased)

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Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 29, 30

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8: 26.9 Integer Partitions: Restricted Number and Part Size

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Table 26.9.1: Partitions

p_{k}\left(n\right)

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$n$	$k$
$n$	…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

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9: 27.2 Functions

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Table 27.2.1: Primes.

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$n$	$p_{n}$	$p_{n+10}$	$p_{n+20}$	$p_{n+30}$	$p_{n+40}$	$p_{n+50}$	$p_{n+60}$	$p_{n+70}$	$p_{n+80}$	$p_{n+90}$
…
5	11	47	97	149	197	257	313	379	439	499
…

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Table 27.2.2: Functions related to division.

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$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
4	2	3	7	17	16	2	18	30	8	8	72	43	42	2	44
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

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10: Bibliography

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M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:: MathReview (J. G. van der Corput), ZentralBlatt
Referenced by:: §30.9(iii).
Encodings:: BibTeX

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S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

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V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

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V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:: In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.
External Links:: Document, MathReview (D. Siersma), ZentralBlatt
Referenced by:: §36.2(i), Ch.36.
Encodings:: BibTeX

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R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.

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External Links:: ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §17.13, §17.13.
Encodings:: BibTeX

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.卡塔尔世界杯奖金_『网址:68707.vip』世界杯竞猜_b5p6v3_2022年11月30日21时49分53秒_3vjzhlfvb.cc

1—10 of 147 matching pages

1: 5.10 Continued Fractions

2: Publications

3: 11 Struve and Related Functions

Chapter 11 Struve and Related Functions

4: 13.30 Tables

5: 24.2 Definitions and Generating Functions

6: 26.2 Basic Definitions

7: Staff

8: 26.9 Integer Partitions: Restricted Number and Part Size

9: 27.2 Functions

10: Bibliography

$n$	$k$
$n$	…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
4	2	3	7	17	16	2	18	30	8	8	72	43	42	2	44
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

$n$	$k$
$n$	…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
4	2	3	7	17	16	2	18	30	8	8	72	43	42	2	44
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

$n$	$k$
$n$	…
6	0	1	4	7	9	10	11	11	11	11	11
7	0	1	4	8	11	13	14	15	15	15	15
…
9	0	1	5	12	18	23	26	28	29	30	30
10	0	1	6	14	23	30	35	38	40	41	42

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
4	2	3	7	17	16	2	18	30	8	8	72	43	42	2	44
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…