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1: 27.16 Cryptography

… ►Applications to cryptography rely on the disparity in computer time required to find large primes and to factor large integers. … ►Procedures for finding such primes require very little computer time. …Messages are coded by a method (described below) that requires only the knowledge of $n$. …With the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer. … ►If $p$ and $q$ are known, $s$ and $y^s$ can be determined (mod $n$) by straightforward calculations that require only a few minutes of machine time. …

2: About MathML

… ►The mathematical notations used in DLMF are complex enough that a fairly complete MathML implementation is required. … ►A good resource for finding and installing math fonts is at Fonts for MathML.

3: Need Help?

… ►

• How do I search within DLMF? See Guide to Searching the DLMF.

• See also the Index or Notations sections.

• Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

4: How to Cite

… ►If you mention a specific equation (or chapter, section, …), you’ll help your readers find it by using our Permalinks & Reference numbers. … ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …

5: 27.19 Methods of Computation: Factorization

… ►Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time. … ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …

6: 29.19 Physical Applications

… ►Macfadyen and Winternitz (1971) finds expansions for the two-body relativistic scattering amplitudes. …

7: 18.40 Methods of Computation

… ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►It is now necessary to take the limit $\epsilon \to 0+$ of $F(x^{\prime }+\mathrm{i}\epsilon )$, and the imaginary part is the required Stieltjes–Perron inversion: …Gautschi (2004, p. 119–120) has explored the $\epsilon \to 0^{+}$ limit via the Wynn $\epsilon$-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in $w(x)$, depending smoothly on $x^{\prime }$, for $N\approx 4000$, for an example involving first numerator Legendre OP’s. … ►Achieving precisions at this level shown above requires higher than normal computational precision, see Gautschi (2009). …

8: 22.20 Methods of Computation

… ►For $x$ real and $k\in (0,1)$, use (22.20.1) with $a_0=1$, $b_0=k^{\prime }\in (0,1)$, $c_0=k$, and continue until $c_N$ is zero to the required accuracy. …If only the value of $\mathrm{dn}(x,k)$ at $x=K$ is required then the exact value is in the table 22.5.1. … ►From the first two terms in (22.10.6) we find $\mathrm{dn}(0.19,\frac{1}{19})=0.999951$. …

9: 23.20 Mathematical Applications

… ►The addition law states that to find the sum of two points, take the third intersection with $C$ of the chord joining them (or the tangent if they coincide); then its reflection in the $x$-axis gives the required sum. …

10: Guide to Searching the DLMF

… ►Therefore, if your query is Ai^2+Bi^2, the system modifies the query so it will find the equations containing the latter expressions. … ►To find more effectively the information you need, especially equations, you may at times wish to specify what you want with descriptive words that characterize the contents but do not occur literally. …