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21: 27.13 Functions

… ►Every even integer $n>4$ is the sum of two odd primes. In this case, $S(n)$ is the number of solutions of the equation $n=p+q$, where $p$ and $q$ are odd primes. …This conjecture dates back to 1742 and was undecided in 2009, although it has been confirmed numerically up to very large numbers. Vinogradov (1937) proves that every sufficiently large odd integer is the sum of three odd primes, and Chen (1966) shows that every sufficiently large even integer is the sum of a prime and a number with no more than two prime factors. … ►Similarly, $G\left(k\right)$ denotes the smallest $m$ for which (27.13.1) has nonnegative integer solutions for all sufficiently large $n$. … ►If $3^k=q{2}^k+r$ with , then equality holds in (27.13.2) provided $r+q\le 2^k$, a condition that is satisfied with at most a finite number of exceptions. …

22: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).

23: 28.4 Fourier Series

… ►Ambiguities in sign are resolved by (28.4.13)–(28.4.16) when $q=0$, and by continuity for the other values of $q$. ►

§28.4(iv) Case $q=0$

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§28.4(v) Change of Sign of $q$

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§28.4(vi) Behavior for Small $q$

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§28.4(vii) Asymptotic Forms for Large $m$

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24: 5.4 Special Values and Extrema

… ►If $p,q$ are integers with , then ►

5.4.19 $$\psi \left(\frac{p}{q}\right)=-\gamma -\ln q-\frac{\pi }{2}\cot \left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum _{k=1}^{q-1}\cos \left(\frac{2\pi kp}{q}\right)\ln \left(2-2\cos \left(\frac{2\pi k}{q}\right)\right).$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, $q$: real or complex variable and $k$: nonnegative integer

Referenced by:

§5.19(i), §5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

…

25: 2.11 Remainder Terms; Stokes Phenomenon

… ►with $m$ a large integer. …with … ►In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable $x$ that is intended to be used. … ►However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►For large $\rho$ the integrand has a saddle point at $t={\mathrm{e}}^{-\mathrm{i}\theta }$. …

26: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

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§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

… ►

§13.8(iii) Large $a$

… ► … ►

§13.8(iv) Large $a$ and $b$

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27: 8.25 Methods of Computation

… ►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the corresponding asymptotic expansions (generally divergent) are used instead. … ►DiDonato and Morris (1986) describes an algorithm for computing $P(a,x)$ and $Q(a,x)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ from the uniform expansions in §8.12. … ►Expansions involving incomplete gamma functions often require the generation of sequences $P(a+n,x)$, $Q(a+n,x)$, or ${\gamma }^{\ast }(a+n,x)$ for fixed $a$ and $n=0,1,2,\mathrm{\dots }$. …

28: 14.15 Uniform Asymptotic Approximations

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§14.15(i) Large $\mu$, Fixed $\nu$

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§14.15(ii) Large $\mu$, $0\le \nu +\frac{1}{2}\le (1-\delta )\mu$

… ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). ►

§14.15(iii) Large $\nu$, Fixed $\mu$

… ►

29: Errata

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18.27.4 $$\sum _{y=0}^N{Q}_n(q^{-y})Q_m(q^{-y}){\left[\genfrac{}{}{0.0pt}{}{N}{y}\right]}_q\frac{{(\alpha q;q)}_y{(\beta q;q)}_{N-y}}{{\left(\alpha q\right)}^y}=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$

We changed the presentation of this equation. Previously the ${\left[\genfrac{}{}{0.0pt}{}{N}{y}\right]}_q\frac{{(\alpha q;q)}_y{(\beta q;q)}_{N-y}}{{\left(\alpha q\right)}^y}$ was presented as $\frac{{(\alpha q,q^{-N};q)}_y{(\alpha \beta q)}^{-y}}{{(q,{\beta }^{-1}{q}^{-N};q)}_y}$.

… ►

An entire new Subsection 13.8(iv) “Large $a$ and $b$”, was added.

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17.11.2 $${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)=\frac{{(b,ax;q)}_{\mathrm{\infty }}}{{(c,x;q)}_{\mathrm{\infty }}}\sum _{n,r\geqq 0}\frac{{(a,b^{\prime };q)}_n{(c/b,x;q)}_r{b}^r{y}^n}{{(q,c^{\prime };q)}_n{(q;q)}_r{(ax;q)}_{n+r}}$$

The factor ${\left(q\right)}_r$ originally used in the denominator has been corrected to be ${(q;q)}_r$.

… ►

The multi-product notation ${(q,c;q)}_m{(q,c^{\prime };q)}_n$ in the denominator of the right-hand side was used.

… ►

A sentence was added recommending §27.14(ii) for properties of ${(q;q)}_{\mathrm{\infty }}$.

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30: 15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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§15.12(ii) Large $c$

… ►For large $b$ and $c$ with $c>b+1$ see López and Pagola (2011). ►

§15.12(iii) Other Large Parameters

… ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).