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## 1—10 of 55 matching pages

##### 1: 34.13 Methods of Computation

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βΊMethods of computation for $\mathit{3}\beta \x81\u2019j$ and $\mathit{6}\beta \x81\u2019j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981).
βΊFor $\mathit{9}\beta \x81\u2019j$ symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989).
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##### 2: 36 Integrals with Coalescing Saddles

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##### 3: 15.17 Mathematical Applications

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βΊIn combinatorics, hypergeometric identities classify single sums of products of binomial coefficients.
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##### 4: 27.16 Cryptography

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βΊWith the most efficient computer techniques devised to date (2010), factoring an 800-digit number may require billions of years on a single computer.
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##### 5: 31.9 Orthogonality

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###### §31.9(i) Single Orthogonality

…##### 6: Notices

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Master Software Index
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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.

##### 7: 3.1 Arithmetics and Error Measures

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###### IEEE Standard

… βΊIn the case of the normalized binary interchange formats, the representation of data for*binary32*(previously*single precision*) ($N=32$, $p=24$, ${E}_{\mathrm{min}}=-126$, ${E}_{\mathrm{max}}=127$),*binary64*(previously*double precision*) ($N=64$, $p=53$, ${E}_{\mathrm{min}}=-1022$, ${E}_{\mathrm{max}}=1023$) and*binary128*(previously*quad precision*) ($N=128$, $p=113$, ${E}_{\mathrm{min}}=-16382$, ${E}_{\mathrm{max}}=16383$) are as in Figure 3.1.1. … βΊ …##### 8: Guide to Searching the DLMF

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Table 3: A sample of recognized symbols
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βΊβΊβΊ

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Symbols | Comments |
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`’` (single quote) |
As a prime, which can stand for the derivative |

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Single-letter terms

##### 9: 26.13 Permutations: Cycle Notation

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βΊA

*transposition*is a permutation that consists of a single cycle of length two. …A permutation that consists of a single cycle of length $k$ can be written as the composition of $k-1$ two-cycles (read from right to left): …##### 10: Bibliography

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Algorithm 609. A portable FORTRAN subroutine for the Bickley functions ${\mathrm{Ki}}_{n}\beta \x81\u2019(x)$
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ACM Trans. Math. Software 9 (4), pp. 480–493.
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Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function.
ACM Trans. Math. Software 9 (4), pp. 494–502.
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Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order.
ACM Trans. Math. Software 12 (3), pp. 265–273.
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Algorithm 556: Exponential integrals.
ACM Trans. Math. Software 6 (3), pp. 420–428.
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Algorithm 588. Fast Hankel transforms using related and lagged convolutions.
ACM Trans. Math. Software 8 (4), pp. 369–370.
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