研究所与大学联盟学士学位证〖办证V信ATV1819〗run

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1: 27.18 Methods of Computation: Primes

… ►It runs in time $O\left({(\ln n)}^{c\ln \ln \ln n}\right)$. … ►That is to say, it runs in time $O\left({(\ln n)}^c\right)$ for some constant $c$. …

2: 27.19 Methods of Computation: Factorization

… ►Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. …

3: About MathML

… ►In rare cases, a browser lacks both MathML support and a robust enough javascript implementation capable of running MathJax; you may wish to visit the Customize DLMF page and choose the HTML+images document format. …

4: 36.15 Methods of Computation

… ►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi }$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. …

5: Philip J. Davis

… ►An interesting anecdote told by Davis reveals that he and mathematician Philip Rabinowitz were dubbed “Heroes of the SEAC” when their Gaussian integration code executed correctly on its first run. …

6: 15.19 Methods of Computation

… ►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.

7: 13.29 Methods of Computation

… ►The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.

8: 22.3 Graphics

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Figure 22.3.26: Density plot of $|\mathrm{sn}(5,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. Grayscale, running from 0 (black) to 10 (white), with $|(\mathrm{sn}(5,k))|>10$ truncated to 10. … Magnify

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Figure 22.3.27: Density plot of $|\mathrm{sn}(10,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. Grayscale, running from 0 (black) to 10 (white), with $|\mathrm{sn}(10,k)|>10$ truncated to 10. … Magnify

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Figure 22.3.28: Density plot of $|\mathrm{sn}(20,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. Grayscale, running from 0 (black) to 10 (white), with $|\mathrm{sn}(20,k)|>10$ truncated to 10. … Magnify

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Figure 22.3.29: Density plot of $|\mathrm{sn}(30,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. Grayscale, running from 0 (black) to 10 (white), with $|\mathrm{sn}(30,k)|>10$ truncated to 10. … Magnify

9: 28.33 Physical Applications

… ►As $\omega$ runs from $0$ to $+\mathrm{\infty }$, with $b$ and $f$ fixed, the point $(q,a)$ moves from $\mathrm{\infty }$ to $0$ along the ray $ℒ$ given by the part of the line $a=(2b/f)q$ that lies in the first quadrant of the $(q,a)$-plane. …

10: DLMF Project News

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