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21: 12.5 Integral Representations

… ►Restrictions on $a$ are not needed in the following two representations: …

22: 18.36 Miscellaneous Polynomials

… ►The restriction to $n\ge 1$ is now apparent: (18.36.7) does not posses a solution if $y(x)$ is a constant. …

23: 19.24 Inequalities

… ►The condition $y\le z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity. …

24: 22.15 Inverse Functions

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25: 28.12 Definitions and Basic Properties

… ►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. …

26: 31.3 Basic Solutions

… ►With similar restrictions to those given in §31.3(i), the following results apply. …

27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Then any self-adjoint extension of $T$ is determined by a linear isometry $U:N_{\mathrm{i}}\to N_{-\mathrm{i}}$ and it is the restriction of $T^{\ast }$ to $\{v+w+Uw\mid v\in 𝒟(T^{\ast \ast }),w\in N_{\mathrm{i}}\}$. … ►Let $n_1,n_1$ be the deficiency indices for $ℒ$ restricted to $(a,c)$, and $n_2,n_2$ the ones for $ℒ$ restricted to $(c,b)$. …Any self-adjoint extension of $ℒ$ can be obtained by restricting ${ℒ}^{\ast }$ to those $f\in 𝒟({ℒ}^{\ast })$ for which, if $n_1=2$, ${ℬ}_1(f)=0$ for a chosen ${ℬ}_1$ at $a$ and, if $n_2=2$, ${ℬ}_2(f)=0$ for a chosen ${ℬ}_2$ at $b$. …

28: 2.1 Definitions and Elementary Properties

… ►But for any given set of coefficients $a_0,a_1,a_2,\mathrm{\dots }$, and suitably restricted $𝐗$ there is an infinity of analytic functions $f(x)$ such that (2.1.14) and (2.1.16) apply. …

29: 5.11 Asymptotic Expansions

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5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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30: 11.2 Definitions

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