limiting values

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21: 22.11 Fourier and Hyperbolic Series

… ►In (22.11.7)–(22.11.12) the left-hand sides are replaced by their limiting values at the poles of the Jacobian functions. …

22: 2.4 Contour Integrals

… ►The branch of ${\omega }_0=\mathrm{ph}\left(p^{\prime \prime }(t_0)\right)$ is the one satisfying $|\theta +2\omega +{\omega }_0|\le \frac{1}{2}\pi$, where $\omega$ is the limiting value of $\mathrm{ph}\left(t-t_0\right)$ as $t\to t_0$ from $b$. …

23: 13.14 Definitions and Basic Properties

… ► ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►Also, unless specified otherwise $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are assumed to have their principal values. ►

§13.14(iii) Limiting Forms as $z\to 0$

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§13.14(iv) Limiting Forms as $z\to \mathrm{\infty }$

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24: 14.3 Definitions and Hypergeometric Representations

… ►When $\mu =m$ ($\in \mathbb{Z}$) (14.3.2) is replaced by its limiting value; see Hobson (1931, §132) for details. …

25: 28.20 Definitions and Basic Properties

… ►When $\nu$ is an integer the right-hand sides of (28.20.25) are replaced by the their limiting values. …

26: 1.4 Calculus of One Variable

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1.4.24 $${⨍}_a^{b}f(x)dx=𝑃{\int }_a^{b}f(x)dx=\underset{ϵ\to 0+}{lim}\left({\int }_a^{c-ϵ}f(x)dx+{\int }_{c+ϵ}^{b}f(x)dx\right),$$

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Defines:

${⨍}_a^b$: Cauchy principal value

Symbols:

$dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.18(vi)

Permalink:

http://dlmf.nist.gov/1.4.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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1.4.25 $${⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=𝑃{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=\underset{b\to \mathrm{\infty }}{lim}{\int }_{-b}^{b}f(x)dx,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral and ${⨍}_a^b$: Cauchy principal value

Referenced by:

§1.14(i)

Permalink:

http://dlmf.nist.gov/1.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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27: Preface

… ►Thus the utilitarian value of the Handbook will be extended far beyond its original scope and the traditional limitations of printed media. …

28: 15.2 Definitions and Analytical Properties

… ►Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …

29: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.16 $$\underset{m\to \mathrm{\infty }}{lim}{\int }_a^b{\left|f(x)-\sum _{n=0}^m{c}_n{\varphi }_n(x)\right|}^{2}dx=0.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.54 $$\underset{ϵ\to 0+}{lim}{\int }_X\frac{f(y)}{x\pm \mathrm{i}ϵ-y}dy=P{⨍}_X\frac{f(y)}{x-y}dy\mp \mathrm{i}\pi f(x).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and ${⨍}_a^b$: Cauchy principal value

Permalink:

http://dlmf.nist.gov/1.18.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

… ► A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …

30: Bibliography N

… ►

Coulomb Functions for Large Values of the Parameter $\eta$ . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.

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Notes:

Rep. CRT-526

External Links:

MathReview (A. Erdélyi)

Referenced by:

§33.7.

Encodings:

BibTeX

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