limiting values

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11: 28.12 Definitions and Basic Properties

… ►However, these functions are not the limiting values of ${\mathrm{me}}_{\pm \nu }(z,q)$ as $\nu \to n$ $(\ne 0)$. … ►Again, the limiting values of ${\mathrm{ce}}_{\nu }(z,q)$ and ${\mathrm{se}}_{\nu }(z,q)$ as $\nu \to n$ $(\ne 0)$ are not the functions ${\mathrm{ce}}_n(z,q)$ and ${\mathrm{se}}_n(z,q)$ defined in §28.2(vi). …

12: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►with limiting value … ►with limiting value …

13: 10.4 Connection Formulas

… ►In (10.4.5), (10.4.7), and (10.4.8) limiting values are taken when $\nu =n$; compare (10.2.3) and (10.2.4). …

14: 4.4 Special Values and Limits

§4.4 Special Values and Limits

… ►

4.4.19 $$\underset{n\to \mathrm{\infty }}{lim}\left(\left(\sum _{k=1}^n\frac{1}{k}\right)-\ln n\right)=\gamma =\mathrm{0.57721\; 56649\; 01532\; 86060}\mathrm{\dots },$$

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Symbols:

$\gamma$: Euler’s constant, $\ln z$: principal branch of logarithm function, $k$: integer and $n$: integer

A&S Ref:

4.1.32 (with 10D value)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Permalink:

http://dlmf.nist.gov/4.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(iii), §4.4 and Ch.4

…

15: 18.6 Symmetry, Special Values, and Limits to Monomials

§18.6 Symmetry, Special Values, and Limits to Monomials

…

16: 10.2 Definitions

… ►When $\nu$ is an integer the right-hand side is replaced by its limiting value: …

17: 8.2 Definitions and Basic Properties

… ►(8.2.9) also holds when $a$ is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. …

18: 10.11 Analytic Continuation

… ►If $\nu =n$ $(\in \mathbb{Z})$, then limiting values are taken in (10.11.2)–(10.11.4): …

19: 10.27 Connection Formulas

… ►When $\nu$ is an integer limiting values are taken: …

20: 20.5 Infinite Products and Related Results

… ►The left-hand sides of (20.5.10) and (20.5.11) are replaced by their limiting values when $\cot z$ or $\tan z$ are undefined. …