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31: 27.2 Functions

… ►An equivalent form states that the $n$th prime $p_n$ (when the primes are listed in increasing order) is asymptotic to $n\ln n$ as $n\to \mathrm{\infty }$: …

32: 27.11 Asymptotic Formulas: Partial Sums

… ►The behavior of a number-theoretic function $f(n)$ for large $n$ is often difficult to determine because the function values can fluctuate considerably as $n$ increases. …

33: 28.29 Definitions and Basic Properties

… ►To every equation (28.29.1), there belong two increasing infinite sequences of real eigenvalues: …

34: Software Index

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An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

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35: 10.21 Zeros

… ►Any positive zero $c$ of the cylinder function ${𝒞}_{\nu }\left(x\right)$ and any positive zero $c^{\prime }$ of ${𝒞}_{\nu }^{\prime }\left(x\right)$ such that $c^{\prime }>|\nu |$ are definable as continuous and increasing functions of $\nu$: … ►In particular, $j_{\nu ,m}$, $y_{\nu ,m}$, $j_{\nu ,m}^{\prime }$, and $y_{\nu ,m}^{\prime }$ are increasing functions of $\nu$ when $\nu \ge 0$. It is also true that the positive zeros $j_{\nu }^{\prime \prime }$ and $j_{\nu }^{\prime \prime \prime }$ of $J_{\nu }^{\prime \prime }\left(x\right)$ and $J_{\nu }^{\prime \prime \prime }\left(x\right)$, respectively, are increasing functions of $\nu$ when $\nu >0$, provided that in the latter case $j_{\nu }^{\prime \prime \prime }>\sqrt{3}$ when . … ►The expansions (10.21.32)–(10.21.39) become progressively weaker as $m$ increases. …

36: 2.11 Remainder Terms; Stokes Phenomenon

… ►The rest of this section is devoted to general methods for increasing this accuracy. … ►We now increase substantially the accuracy of (2.11.6) in $|\mathrm{ph}z|\le \pi$ by re-expanding the remainder term. …

37: 3.11 Approximation Techniques

… ►Since $L_0=1$, $L_n$ is a monotonically increasing function of $n$, and (for example) $L_{1000}=4.07\mathrm{\dots }$, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … ►By taking more derivatives into account, the smoothness of the spline will increase. …

38: 8.11 Asymptotic Approximations and Expansions

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8.11.12 $$\mathrm{\Gamma }(z,z)\sim z^{z-1}{\mathrm{e}}^{-z}\left(\sqrt{\frac{\pi }{2}}{z}^{\frac{1}{2}}-\frac{1}{3}+\frac{\sqrt{2\pi }}{24z^{\frac{1}{2}}}-\frac{4}{135z}+\frac{\sqrt{2\pi }}{576z^{\frac{3}{2}}}+\frac{8}{2835z^2}+\mathrm{\dots }\right),$$ $|\mathrm{ph}z|\le 2\pi -\delta$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

6.5.35 (Modified form)

Referenced by:

§8.11(v), Firefox 3 users should upgrade, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

The sector of validity for the above equation was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$.

See also:

Annotations for §8.11(v), §8.11 and Ch.8

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39: 22.19 Physical Applications

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Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.5,0.9999,1.0001,2$. When , $\mathrm{am}(x,k)$ increases monotonically indicating that the motion of the pendulum is unbounded in $\theta$, corresponding to free rotation about the fulcrum; compare Figure 22.16.1. … Magnify

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40: 28.6 Expansions for Small $q$

… ►It is conjectured that for large $n$, the radii increase in proportion to the square of the eigenvalue number $n$; see Meixner et al. (1980, §2.4). …