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11: 9.10 Integrals

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§9.10(viii) Repeated Integrals

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12: Software Index

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	Open Source													With Book						Commercial
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7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
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13: 2.6 Distributional Methods

… ►To assign a distribution to the function $f_n(t)$, we first let $f_{n,n}(t)$ denote the $n$th repeated integral (§1.4(v)) of $f_n$: ►

2.6.15 $$f_{n,n}(t)=\frac{{(-1)}^n}{(n-1)!}{\int }_t^{\mathrm{\infty }}{(\tau -t)}^{n-1}{f}_n(\tau )d\tau .$$

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Defines:

$f_{n,n}(t)$: $n$th repeated integral (locally)

Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer and $f_n(t)$: remainder

Referenced by:

§2.6(ii), §2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

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2.6.16 $$⟨f_n,\varphi ⟩={(-1)}^n{\int }_0^{\mathrm{\infty }}{f}_{n,n}(t){\varphi }^{(n)}(t)dt,$$ $\varphi \in 𝒯$,

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Symbols:

$dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral, $n$: nonnegative integer, $f_n(t)$: remainder, $𝒯$: space of decreasing functions and $f_{n,n}(t)$: $n$th repeated integral

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

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2.6.26 $$\underset{\epsilon \to 0}{lim}⟨f_n,{\varphi }_{\epsilon }⟩=n!{\int }_0^{\mathrm{\infty }}\frac{f_{n,n}(t)}{{(t+z)}^{n+1}}dt.$$

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Symbols:

$dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $!$: factorial (as in $n!$), $\int$: integral, $\epsilon$: small positive number, $n$: nonnegative integer, $f_n(t)$: remainder and $f_{n,n}(t)$: $n$th repeated integral

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

… ► $f_{n,j}(t)$ being the $j$th repeated integral of $f_n$; compare (2.6.15). …

14: 36.7 Zeros

… ►Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …

15: 36.10 Differential Equations

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16: 1.4 Calculus of One Variable

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Repeated Integrals

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17: 1.5 Calculus of Two or More Variables

… ►where the right-hand side is interpreted as the repeated integral … ►In the cases (1.5.30) and (1.5.33) they are defined by taking limits in the repeated integrals (1.5.32) and (1.5.34) in an analogous manner to (1.4.22)–(1.4.23). … ►whenever both repeated integrals exist and at least one is absolutely convergent. …

18: Bibliography H

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Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:

ZentralBlatt

Referenced by:

§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

Encodings:

BibTeX

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19: 25.2 Definition and Expansions

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25.2.9 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.2.8) by repeated integration by parts.

Referenced by:

§25.18(i), (25.2.10)

Permalink:

http://dlmf.nist.gov/25.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

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20: 1.6 Vectors and Vector-Valued Functions

… ►In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. …