���������������������������������V���ATV1819���e3y6

SearchAdvancedHelp

(0.000 seconds)

31—40 of 69 matching pages

31: 7.21 Physical Applications

… ►Voigt functions $𝖴(x,t)$, $𝖵(x,t)$, can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. …

32: 12.11 Zeros

… ►If $a>-\frac{1}{2}$, then $V(a,x)$ has no positive real zeros, and if $a=\frac{3}{2}-2n$, $n\in \mathbb{Z}$, then $V(a,x)$ has a zero at $x=0$. … ►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

33: 28.33 Physical Applications

… ►with $W(x,y,t)={\mathrm{e}}^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with $k^2={\omega }^2\rho /\tau$. …The separated solutions $V_n(\xi ,\eta )$ must be $2\pi$-periodic in $\eta$, and have the form ►

28.33.2 $$V_n(\xi ,\eta )=\left(c_n{\mathrm{M}}_n^{(1)}(\xi ,\sqrt{q})+d_n{\mathrm{M}}_n^{(2)}(\xi ,\sqrt{q})\right){\mathrm{me}}_n(\eta ,q),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $q=h^2$: parameter, $n$: integer, $V(\xi ,\eta )$: solution, $d_n$: constants, $\eta$: variable, $\xi$: variable and $c_n$: constants

Permalink:

http://dlmf.nist.gov/28.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.33(ii), §28.33 and Ch.28

…

34: 18.39 Applications in the Physical Sciences

… ►The properties of $V(x)$ determine whether the spectrum, this being the set of eigenvalues of $\mathscr{H}$, is discrete, continuous, or mixed, see §1.18. … ►where $V(x)$ is assumed to be independent of time. … ►Now use spherical coordinates (1.5.16) with $r$ instead of $\rho$, and assume the potential $V$ to be radial. Then write $V(r)$ instead of $V(𝐱)$. … ►Analogous to (18.39.8) the 3D time-independent Schrödinger equation with potential $V(r)$ is …

35: 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. …

36: 18.9 Recurrence Relations and Derivatives

… ►

Table 18.9.2: Classical OP’s: recurrence relations (18.9.2_1).

► ►►►

$p_n(x)$	$a_n$	$b_n$	$c_n$
…
$V_n\left(x\right)$	$\frac{1}{2}$	$-\frac{1}{2}{\delta }_{n,0}$	$\frac{1}{2}$
…

►

… ►

18.9.11 $$V_n\left(x\right)+V_{n-1}\left(x\right)=2T_n\left(x\right),$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.9.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the left-hand side we replaced $W_n\left(x\right)+W_{n-1}\left(x\right)$ with $V_n\left(x\right)+V_{n-1}\left(x\right)$. For further details see Errata.

See also:

Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

►

18.9.12 $$T_{n+1}\left(x\right)+T_n\left(x\right)=(1+x)V_n\left(x\right).$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.7(i), §18.9(ii), §18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.9.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $(1+x)W_n\left(x\right)$ with $(1+x)V_n\left(x\right)$. For further details see Errata.

See also:

Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

…

37: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.11 $$|R_{\mathrm{\ell }}(\nu ,z)|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\mathrm{\infty }}\left(t^{-1}\right)\right),$$

ⓘ

Defines:

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

Symbols:

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

Permalink:

http://dlmf.nist.gov/10.40.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

►where $𝒱$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\mathrm{\Re }t|$ changes monotonically. Bounds for ${𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by ►

10.40.12

ⓘ

Symbols:

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

Referenced by:

§10.40(i), §10.40(iii), Erratum (V1.0.11) for Equation (10.40.12)

Permalink:

http://dlmf.nist.gov/10.40.E12

Encodings:

pMML, png, TeX

Correction (effective with 1.0.11):

Originally the third constraint was written incorrectly as $\pi \le |\mathrm{ph}z|\le \frac{3}{2}\pi$.

Suggested 2014-11-05 by Gergő Nemes

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

…

38: 18.3 Definitions

… ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

► ►►►

Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…
Chebyshev of third kind	$V_n\left(x\right)$	$(-1,1)$	${(1-x)}^{-\frac{1}{2}}{(1+x)}^{\frac{1}{2}}$	$\pi$	$2^n$	$-\frac{1}{2}$
…

►

…

39: 1.6 Vectors and Vector-Valued Functions

… ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. … ►

1.6.58 $${\iiint }_V(\nabla \cdot 𝐅)dV={\iint }_S𝐅\cdot d𝐒,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $S$: parameterized surface and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

… ►

1.6.59 $${\iiint }_V(f{\nabla }^{2}g+\nabla f\cdot \nabla g)dV={\iint }_{S}f\frac{\partial g}{\partial n}dA,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E59

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

… ►

1.6.60 $${\iiint }_V(f{\nabla }^{2}g-g{\nabla }^{2}f)dV={\iint }_S\left(f\frac{\partial g}{\partial n}-g\frac{\partial f}{\partial n}\right)dA,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $S$: parameterized surface, $A(S)$: area of a parameterized smooth surface $S$ and $V$: closed region

Permalink:

http://dlmf.nist.gov/1.6.E60

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(v), §1.6(v), §1.6 and Ch.1

►where $\partial g/\partial n=\nabla g\cdot 𝐧$ is the derivative of $g$ normal to the surface outwards from $V$ and $𝐧$ is the unit outer normal vector. …

40: 9.9 Zeros

… ►

9.9.7 $${\mathrm{Ai}}^{\prime }\left(a_k\right)={(-1)}^{k-1}V\left(\frac{3}{8}\pi (4k-1)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $a_k$: $k$th zero of Airy $\mathrm{Ai}$ , $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 20))

Permalink:

http://dlmf.nist.gov/9.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

… ►

9.9.11 $${\mathrm{Bi}}^{\prime }\left(b_k\right)={(-1)}^{k-1}V\left(\frac{3}{8}\pi (4k-3)\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $b_k$: $k$th zero of Airy $\mathrm{Bi}$ , $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 21))

Permalink:

http://dlmf.nist.gov/9.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

… ►

9.9.15 $${\mathrm{Bi}}^{\prime }\left({\beta }_k\right)={(-1)}^k\sqrt{2}{\mathrm{e}}^{-\pi \mathrm{i}/6}V\left(\frac{3}{8}\pi (4k-1)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k$: $k$th complex zero of Airy $\mathrm{Bi}$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

… ►

9.9.20 $$V(t)\sim {\pi }^{-1/2}{t}^{1/6}\left(1+\frac{5}{48}{t}^{-2}-\frac{1525}{4608}{t}^{-4}+\frac{\mathrm{23\hspace{0.33em}97875}}{\mathrm{6\hspace{0.33em}63552}}{t}^{-6}-\frac{\mathrm{7\hspace{0.33em}48989\hspace{0.33em}40625}}{\mathrm{8918\hspace{0.33em}13888}}{t}^{-8}+\frac{\mathrm{14419\hspace{0.33em}83037\hspace{0.33em}34375}}{\mathrm{4\hspace{0.33em}28070\hspace{0.33em}66624}}{t}^{-10}-\mathrm{\cdots }\right),$$

ⓘ

Defines:

$V$: expansion (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion and $\pi$: the ratio of the circumference of a circle to its diameter

Source:

Olver (1954, (A 19))

Permalink:

http://dlmf.nist.gov/9.9.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

…