large b
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31—40 of 94 matching pages
31: 27.16 Cryptography
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►Applications to cryptography rely on the disparity in computer time required to find large primes and to factor large integers.
►For example, a code maker chooses two large primes and of about 400 decimal digits each.
…For this reason, the codes are considered unbreakable, at least with the current state of knowledge on factoring large numbers.
►To code a message by this method, we replace each letter by two digits, say , , , , and divide the message into pieces of convenient length smaller than the public value .
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32: 10.17 Asymptotic Expansions for Large Argument
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10.17.12
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33: 24.16 Generalizations
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►For extensions of to complex values of , , and , and also for uniform asymptotic expansions for large
and large
, see Temme (1995b) and López and Temme (1999b, 2010b).
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34: Preface
35: 6.18 Methods of Computation
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, , and can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values , say, where is an arbitrary large integer, and normalizing via .
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36: 10.41 Asymptotic Expansions for Large Order
§10.41 Asymptotic Expansions for Large Order
►§10.41(i) Asymptotic Forms
… ►§10.41(ii) Uniform Expansions for Real Variable
… ► … ►37: 10.74 Methods of Computation
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►If or is large compared with , then the asymptotic expansions of §§10.17(i)–10.17(iv) are available.
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►For large positive real values of the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used.
Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large
or , whether or not is large.
It should be noted, however, that there is a difficulty in evaluating the coefficients , , , and , from the explicit expressions (10.20.10)–(10.20.13) when is close to owing to severe cancellation.
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►And since there are no error terms they could, in theory, be used for all values of ; however, there may be severe cancellation when is not large compared with .
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38: 25.11 Hurwitz Zeta Function
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25.11.6
, , .
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►For see §24.2(iii).
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25.11.19
, , .
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§25.11(xii) -Asymptotic Behavior
… ►Similarly, as in the sector , …39: Staff
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Adri B. Olde Daalhuis, Mathematics Editor, The University of Edinburgh
Ronald F. Boisvert, Editor at Large, NIST
Vadim B. Kuznetsov, University of Leeds, Chap. 31
Adri B. Olde Daalhuis, University of Edinburgh, Chaps. 13, 15, 16
Richard B. Paris, University of Abertay, Chaps. 8, 11
40: 28.35 Tables
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National Bureau of Standards (1967) includes the eigenvalues , for with , and with ; Fourier coefficients for and for , , respectively, and various values of in the interval ; joining factors , for with (but in a different notation). Also, eigenvalues for large values of . Precision is generally 8D.