biharmonic operator

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11: Guide to Searching the DLMF

… ►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: … ►

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adj, prec/n, and near/n, where n is any positive natural number.

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12: About MathML

… ►As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. …

13: 18.1 Notation

… ►

$x$-Differences

►Forward differences: … ►Backward differences: … ►Central differences in imaginary direction: … ►In Koekoek et al. (2010) ${\delta }_x$ denotes the operator $\mathrm{i}{\delta }_x$.

14: 31.17 Physical Applications

… ►We use vector notation $[𝐬,𝐭,𝐮]$ (respective scalar $(s,t,u)$) for any one of the three spin operators (respective spin values). … ►

$${𝐻}_s\mathrm{\Psi }(𝐱)\equiv (-2𝐬\cdot 𝐭-(2/a)𝐬\cdot 𝐮)\mathrm{\Psi }(𝐱)=h_s\mathrm{\Psi }(𝐱),$$

… ►The operators ${𝐉}^2$ and ${𝐻}_s$ admit separation of variables in $z_1,z_2$, leading to the following factorization of the eigenfunction $\mathrm{\Psi }(𝐱)$: …

15: William P. Reinhardt

… ►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …

16: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(iv) Matrices as Linear Operators

►

Linear Operators in Finite Dimensional Vector Spaces

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Self-Adjoint Operators on ${𝐄}_n$

… ►Real symmetric ($𝐀={𝐀}^{\mathrm{T}}$) and Hermitian ($𝐀={𝐀}^{\mathrm{H}}$) matrices are self-adjoint operators on ${𝐄}_n$. …

17: 14.30 Spherical and Spheroidal Harmonics

… ►

Parity Operation

… ►Here, in spherical coordinates, $L^2$ is the squared angular momentum operator: ►

14.30.12 $$L^2=-{\mathrm{\hslash }}^2\left(\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{{\partial \varphi }^2}\right),$$

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Defines:

$L$: angular momentum operator (locally)

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\sin z$: sine function

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/14.30.E12

Encodings:

pMML, png, TeX

See also:

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

►and ${\mathrm{L}}_z$ is the $z$ component of the angular momentum operator ►

14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

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Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

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

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18: 1.15 Summability Methods

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1.15.47 $${𝐼}^{\alpha }f(x)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^x{(x-t)}^{\alpha -1}f(t)dt.$$

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Defines:

${𝐼}^{\alpha }$: fractional integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.15(vi), §1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)

Permalink:

http://dlmf.nist.gov/1.15.E47

Encodings:

pMML, png, TeX

See also:

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

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1.15.48 $${𝐼}^{\alpha }{𝐼}^{\beta }={𝐼}^{\alpha +\beta },$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\beta >0$.

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Symbols:

$\mathrm{\Re }$: real part and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vi)

Permalink:

http://dlmf.nist.gov/1.15.E48

Encodings:

pMML, png, TeX

See also:

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

… ►

1.15.50 $${𝐼}^{\alpha }f(x)=\sum _{k=0}^{\mathrm{\infty }}\frac{k!}{\mathrm{\Gamma }\left(k+\alpha +1\right)}{a}_k{x}^{k+\alpha }.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii)

Permalink:

http://dlmf.nist.gov/1.15.E50

Encodings:

pMML, png, TeX

See also:

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

… ►

1.15.51 $${𝐷}^{\alpha }f(x)=\frac{d^n}{{dx}^n}{𝐼}^{n-\alpha }f(x),$$

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Defines:

${𝐷}^{\alpha }$: fractional derivative (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E51

Encodings:

pMML, png, TeX

See also:

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

… ►

1.15.52 $${𝐷}^k{𝐼}^{\alpha }={𝐷}^n{𝐼}^{\alpha +n-k},$$ $k=1,2,\mathrm{\dots },n$.

ⓘ

Symbols:

$k$: integer, $n$: nonnegative integer, ${𝐼}^{\alpha }$: fractional integral and ${𝐷}^{\alpha }$: fractional derivative

Referenced by:

§1.15(vii)

Permalink:

http://dlmf.nist.gov/1.15.E52

Encodings:

pMML, png, TeX

See also:

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

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19: 1.1 Special Notation

… ► ►►►

$x,y$	real variables.
…
$ℒ$	linear operator defined on a manifold $ℳ$
…

…

20: 10.17 Asymptotic Expansions for Large Argument

… ►

10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right),$$

ⓘ

Defines:

$R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.17(iv), Erratum (V1.0.10) for Equation (10.17.14)

Permalink:

http://dlmf.nist.gov/10.17.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

►where $𝒱$ denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that $|\mathrm{\Im }t|$ changes monotonically. Bounds for ${𝒱}_{z,\mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by ►

10.17.15

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable and $\chi (\mathrm{\ell })$

Referenced by:

§10.17(iv)

Permalink:

http://dlmf.nist.gov/10.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

… ►The bounds (10.17.15) also apply to ${𝒱}_{z,-\mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ in the conjugate sectors. …