repeated integrals of

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11: 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

… ►

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

… ►

2.6.47 $${\delta }_n(x)=\sum _{j=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)\frac{\mathrm{\Gamma }\left(\mu +1\right)}{\mathrm{\Gamma }\left(\mu +1-j\right)}{𝐼}^{\mu }\left(t^{n-j}{f}_{n,j}\right)(x),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mu$: order, ${𝐼}^{\mu }$: fractional integral , $n$: nonnegative integer, ${\delta }_n(x)$: sum and $f_{n,n}(t)$: $n$th repeated integral

Permalink:

http://dlmf.nist.gov/2.6.E47

Encodings:

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See also:

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

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

12: 1.4 Calculus of One Variable

… ►

Repeated Integrals

…

13: 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. …

14: Bibliography H

… ►

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

…

15: Bibliography

… ►

Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

…

16: 10.22 Integrals

… ►When $\alpha =m=1,2,3,\mathrm{\dots }$ the left-hand side of (10.22.36) is the $m$th repeated integral of $J_{\nu }\left(x\right)$ (§§1.4(v) and 1.15(vi)). …

17: 25.2 Definition and Expansions

… ►

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

…

18: 19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). …

19: 36.10 Differential Equations

§36.10 Differential Equations

►

§36.10(i) Equations for ${\mathrm{\Psi }}_K\left(𝐱\right)$

… ► $K=2$, cusp: … ► $K=3$, swallowtail: … ►In terms of the normal forms (36.2.2) and (36.2.3), the ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$ satisfy the following operator equations …

20: 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 …