integral over

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1: Bibliography I

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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2: 1.6 Vectors and Vector-Valued Functions

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§1.6(v) Surfaces and Integrals over Surfaces

… ►The integral of a continuous function $f(x,y,z)$ over a surface $S$ is …

3: 11.6 Asymptotic Expansions

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11.6.3 $${\int }_0^z{𝐊}_0\left(t\right)dt-\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.32

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E3

Encodings:

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See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.4 $${\int }_0^z{𝐌}_0\left(t\right)dt+\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.8

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E4

Encodings:

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See also:

Annotations for §11.6(i), §11.6 and Ch.11

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4: 10.43 Integrals

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§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

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§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

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5: 36.9 Integral Identities

… ►For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). …

6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.13 $$c_n=⟨f,{\varphi }_n⟩={\int }_a^{b}f(x)\overline{{\varphi }_n(x)}dx,$$ $f\in L^2\left(X\right)$,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E13

Encodings:

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See also:

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

… ►The adjoint $T^{\ast }$ of $T$ does satisfy $⟨Tf,g⟩=⟨f,T^{\ast }g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. … ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. … ►

1.18.51 $$⟨F(T)f,f⟩={\int }_0^{\mathrm{\infty }}F(\lambda ){\left|\widehat{f}(\lambda )\right|}^{2}d\lambda .$$

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Symbols:

$dx$: differential of $x$ , $⟨f,g⟩$: inner product over functions, $\int$: integral and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E51

Encodings:

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See also:

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

… ►In what follows, integrals over the continuous parts of the spectrum will be denoted by ${𝝈}_c$, and sums over the discrete spectrum by ${𝝈}_p$, with $𝝈={𝝈}_c\cup {𝝈}_p$ denoting the full spectrum. …

7: 10.22 Integrals

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§10.22(ii) Integrals over Finite Intervals

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§10.22(iii) Integrals over the Interval $(x,\mathrm{\infty })$

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§10.22(iv) Integrals over the Interval $(0,\mathrm{\infty })$

… ►Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

8: 19.29 Reduction of General Elliptic Integrals

… ►These theorems reduce integrals over a real interval $(y,x)$ of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over $(0,\mathrm{\infty })$ containing the square root of a cubic polynomial (compare §19.16(i)). …

9: 14.30 Spherical and Spheroidal Harmonics

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14.30.8 $${\int }_0^{2\pi }{\int }_0^{\pi }\overline{Y_{l_1,m_1}(\theta ,\varphi )}{Y}_{l_2,m_2}(\theta ,\varphi )\sin \theta d\theta d\varphi ={\delta }_{l_1,l_2}{\delta }_{m_1,m_2}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E8

Encodings:

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Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the integralover $\theta$ been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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10: Bibliography M

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On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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External Links:

ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

Encodings:

BibTeX

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