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1: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

2: Bibliography B

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Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:

ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§15.8(v), §20.11(iii).

Encodings:

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Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-66170-6; 0-521-66646-5, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§1.3(ii), §26.10(ii), §26.10(iii), §26.12(i), §26.12(ii), §26.12(iii), §26.19, §26.20, §26.9(iii), Profile David M. Bressoud.

Encodings:

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

Encodings:

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3: Peter L. Walker

… ►Walker is now retired and living in Cheltenham, UK. …

4: William P. Reinhardt

… ►Reinhardt was elected to Fellowship in: The American Physical Society 1980, the American Association for the Advancement of Science 1983, Phi Beta Kappa 1998, and The Institute of Physics (UK) 2000. …

5: Errata

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18.2.5

$h_n={\displaystyle {\int }_a^b}{\left(p_n(x)\right)}^{2}w(x)dx$

$\text{ or }{h}_n={\displaystyle \sum _{x\in X}}{\left(p_n(x)\right)}^2{w}_x$

$\text{ or }{h}_n={\displaystyle {\int }_a^b}{\left(p_n(x)\right)}^{2}d\mu (x)$

18.2.6

${\tilde{h}}_n={\displaystyle {\int }_a^b}x{\left(p_n(x)\right)}^{2}w(x)dx$

$\text{ or }{\tilde{h}}_n={\displaystyle \sum _{x\in X}}x{\left(p_n(x)\right)}^2{w}_x$

$\text{ or }{\tilde{h}}_n={\displaystyle {\int }_a^b}x{\left(p_n(x)\right)}^{2}d\mu (x)$

The third alternatives, involving $d\mu (x)$, were included.

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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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A sentence was added to inform the reader of alternatives to Steed’s algorithm, namely the Lentz algorithm (see e.g., Lentz (1976)) and the modified Lentz algorithm (see e.g., Thompson and Barnett (1986)).

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6: Publications

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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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7: Bibliography W

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Lectures on Quantum Mechanics. Cambridge University Press, Cambridge, UK.

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External Links:

ISBN 978-1-107-02872-2

Referenced by:

§18.39(ii).

Encodings:

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8: Bibliography S

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Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.

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Notes:

Available from the library, University of Surrey, UK

External Links:

LibraryRecord

Referenced by:

§31.9(i), §31.9(ii), Ch.31, Profile Brian D. Sleeman.

Encodings:

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Elliptic Solitons and Heun’s Equation. In The Kowalevski Property (Leeds, UK, 2000), V. B. Kuznetsov (Ed.), CRM Proc. Lecture Notes, Vol. 32, pp. 287–306.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.8.

Encodings:

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9: 7.1 Special Notation

… ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. …

10: 5.1 Special Notation

… ►Alternative notations for this function are: $\mathrm{\Pi }(z-1)$ (Gauss) and $(z-1)!$. Alternative notations for the psi function are: $\mathrm{\Psi }(z-1)$ (Gauss) Jahnke and Emde (1945); $\mathrm{\Psi }(z)$ Davis (1933); $𝖥(z-1)$ Pairman (1919). …