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11: 10.71 Integrals

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§10.71(i) Indefinite Integrals

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§10.71(ii) Definite Integrals

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§10.71(iii) Compendia

►For infinite double integrals involving Kelvin functions see Prudnikov et al. (1986b, pp. 630–631). … ►

12: 18.18 Sums

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Laguerre

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13: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

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14: Bibliography B

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Integral transforms with generalized Legendre functions as kernels. Compositio Math. 18, pp. 235–287.

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External Links:

MathReview (R. K. Saxena), ZentralBlatt

Referenced by:

§14.20(vi), §14.29, §14.31(ii).

Encodings:

BibTeX

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15: 13.27 Mathematical Applications

… ►The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. …

16: 10.73 Physical Applications

… ►The analysis of the current distribution in circular conductors leads to the Kelvin functions $\mathrm{ber}x$, $\mathrm{bei}x$, $\ker x$, and $\mathrm{kei}x$. …

17: 1.15 Summability Methods

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1.15.12 $$P(r,\theta )=\frac{1-r^2}{1-2r\cos \theta +r^2}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{r}^{\left|n\right|}{\mathrm{e}}^{\mathrm{i}n\theta },$$ ,

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Defines:

$P(r,\theta )$: Poisson kernel (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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1.15.15 $$K_n(\theta )=\frac{1}{n+1}{\left(\frac{\sin \left(\frac{1}{2}(n+1)\theta \right)}{\sin \left(\frac{1}{2}\theta \right)}\right)}^2,$$

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Defines:

$K_n(\theta )$: Fejér kernel (locally)

Symbols:

$\sin z$: sine function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.15.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(iii), §1.15(iii), §1.15 and Ch.1

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1.15.37 $$h(x,y)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)P(x-t,y)dt$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $P(x,y)$: Poisson kernel and $h(x,y)$: function

Permalink:

http://dlmf.nist.gov/1.15.E37

Encodings:

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Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

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1.15.41 $$K_R(s)=\frac{1}{\pi R}\frac{1-\cos \left(Rs\right)}{s^2},$$

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Defines:

$K_R(s)$: Fejér kernel (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\cos z$: cosine function

Permalink:

http://dlmf.nist.gov/1.15.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

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1.15.45 $${\sigma }_R(\theta )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)K_R(\theta -t)dt.$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $K_R(s)$: Fejér kernel and ${\sigma }_R(\theta )$: function

Permalink:

http://dlmf.nist.gov/1.15.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

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18: Bibliography W

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On a generalization of the functions $\mathrm{ber}$ x, $\mathrm{bei}$ x, $\ker$ x, $\mathrm{kei}$ x. Quart. J. Pure Appl. Math. 42, pp. 316–342.

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External Links:

ZentralBlatt

Referenced by:

§10.61(iii), §10.61(iv), §10.65(i), §10.65(ii), §10.65(iii), §10.68(iii).

Encodings:

BibTeX

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19: Bibliography M

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Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.10(ii).

Encodings:

BibTeX

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20: 20.13 Physical Applications

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …