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1: 25.10 Zeros

… ►Calculations relating to the zeros on the critical line make use of the real-valued function …where … ►Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp \left(\mathrm{i}\vartheta (t)\right)$ to obtain the Riemann–Siegel formula: …where $R(t)=O\left(t^{-1/4}\right)$ as $t\to \mathrm{\infty }$. … ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

2: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

3: Sidebar 7.SB1: Diffraction from a Straightedge

… ►The intensity distribution follows ${|ℱ\left(x\right)|}^2$, where $ℱ$ is the Fresnel integral (See 7.3.4). …The faint circular patterns are additional diffraction effects due to imperfections in the edge.

4: Stephen M. Watt

… ►Cheriton School of Computer Science at the University of Waterloo, where from 2015 to 2020 he also served as Dean of the Faculty of Mathematics. … ►Prior to joining the University of Waterloo, Watt was Distinguished University Professor of the University of Western Ontario and Professor at the University of Nice-Sophia Antipolis. …

5: Joyce E. Conlon

… ►In 1999 she joined the NIST Mathematical and Computational Sciences Division, where she contributed to the DLMF project, especially in the construction of the bibliography. …

6: 26.2 Basic Definitions

… ►A permutation is a one-to-one and onto function from a non-empty set to itself. If the set consists of the integers 1 through $n$, a permutation $\sigma$ can be thought of as a rearrangement of these integers where the integer in position $j$ is $\sigma (j)$. … ►Given a finite set $S$ with permutation $\sigma$, a cycle is an ordered equivalence class of elements of $S$ where $j$ is equivalent to $k$ if there exists an $\mathrm{\ell }=\mathrm{\ell }(j,k)$ such that $j={\sigma }^{\mathrm{\ell }}(k)$, where ${\sigma }^1=\sigma$ and ${\sigma }^{\mathrm{\ell }}$ is the composition of $\sigma$ with ${\sigma }^{\mathrm{\ell }-1}$. …The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. … ►The integers whose sum is $n$ are referred to as the parts in the partition. …

7: Notices

… ►Pursuant to Title 17 USC 105, the National Institute of Standards and Technology (NIST), United States Department of Commerce, is authorized to receive and hold copyrights transferred to it by assignment or otherwise. … ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. …Thus, we seek to provide DLMF users with links to sources of such software. … ►

Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

►

In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

…

8: 5.10 Continued Fractions

… ►

5.10.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)+z-\left(z-\frac{1}{2}\right)\ln z-\frac{1}{2}\ln \left(2\pi \right)=\frac{a_0}{z+}\frac{a_1}{z+}\frac{a_2}{z+}\frac{a_3}{z+}\frac{a_4}{z+}\frac{a_5}{z+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $z$: complex variable and $a_k$: coefficient

Referenced by:

Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.10.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.10 and Ch.5

►where …For exact values of $a_7$ to $a_{11}$ and 40S values of $a_0$ to $a_{40}$, see Char (1980). …

9: 8.26 Tables

… ►

Pearson (1965) tabulates the function $I(u,p)$ ($=P(p+1,u)$) for $p=-1(.05)0(.1)5(.2)50$, $u=0(.1)u_p$ to 7D, where $I(u,u_p)$ rounds off to 1 to 7D; also $I(u,p)$ for $p=-0.75(.01)-1$, $u=0(.1)6$ to 5D.

►

Zhang and Jin (1996, Table 3.8) tabulates $\gamma (a,x)$ for $a=0.5,1,3,5,10,25,50,100$, $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

… ►

Pearson (1968) tabulates $I_x(a,b)$ for $x=0.01(.01)1$, $a,b=0.5(.5)11(1)50$, with $b\le a$, to 7D.

… ►

Stankiewicz (1968) tabulates $E_n\left(x\right)$ for $n=1(1)10$, $x=0.01(.01)5$ to 7D.

►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

10: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers $a_1(mod{m}_1)$, $a_2(mod{m}_2)$, $a_3(mod{m}_3)$, and $a_4(mod{m}_4)$, where each $a_j$ has no more than five digits. These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …