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1: 12.8 Recurrence Relations and Derivatives

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12.8.5 $$zV(a,z)-V(a+1,z)+(a-\frac{1}{2})V(a-1,z)=0,$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.8

Permalink:

http://dlmf.nist.gov/12.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.6 $$V^{\prime }(a,z)-\frac{1}{2}zV(a,z)-(a-\frac{1}{2})V(a-1,z)=0,$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.5

Permalink:

http://dlmf.nist.gov/12.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.7 $$V^{\prime }(a,z)+\frac{1}{2}zV(a,z)-V(a+1,z)=0,$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.6

Permalink:

http://dlmf.nist.gov/12.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.8 $$2V^{\prime }(a,z)-V(a+1,z)-(a-\frac{1}{2})V(a-1,z)=0.$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.7

Permalink:

http://dlmf.nist.gov/12.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.11 $$\frac{d^m}{{dz}^m}\left({\mathrm{e}}^{\frac{1}{4}{z}^2}V(a,z)\right)={\mathrm{e}}^{\frac{1}{4}{z}^2}V(a+m,z),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§12.8(ii)

Permalink:

http://dlmf.nist.gov/12.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(ii), §12.8 and Ch.12

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2: 7.19 Voigt Functions

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Figure 7.19.2: Voigt function $𝖵(x,t)$, $t=0.1$, $2.5$, $5$, $10$. Magnify

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$\underset{t\to 0}{lim}𝖵(x,t)={\displaystyle \frac{x}{1+x^2}}.$

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$𝖵(-x,t)=-𝖵(x,t).$

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$-1\le 𝖵(x,t)\le 1.$

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7.19.9 $$𝖴(x,t)=1-x𝖵(x,t)-2t\frac{\partial 𝖵(x,t)}{\partial x}.$$

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(iii), §7.19 and Ch.7

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3: 5.19 Mathematical Applications

… ►The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by ►

$V={\displaystyle \frac{{\pi }^{\frac{1}{2}n}{r}^n}{\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}},$

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$S={\displaystyle \frac{2{\pi }^{\frac{1}{2}n}{r}^{n-1}}{\mathrm{\Gamma }\left(\frac{1}{2}n\right)}}={\displaystyle \frac{n}{r}}V.$

4: 12.3 Graphics

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Figure 12.3.2: $V(a,x)$, $a$ = 0. … Magnify

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Figure 12.3.4: $V(a,x)$, $a$ = $-0.5$, $-2$, $-3.5$, $-5$. Magnify

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Figure 12.3.8: $V(a,x)$, $-2.5\le a\le 2.5$, $-2.5\le x\le 2.5$. Magnify 3D Help

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5: 36.13 Kelvin’s Ship-Wave Pattern

… ►A ship moving with constant speed $V$ on deep water generates a surface gravity wave. … ►

36.13.2 $$\rho =gr/V^2.$$

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Symbols:

$V$: speed and $g$: gravity

Permalink:

http://dlmf.nist.gov/36.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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36.13.6 $$\omega (𝐤)=\sqrt{gk}+𝐕\cdot 𝐤.$$

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Symbols:

$k$: variable and $g$: gravity

Permalink:

http://dlmf.nist.gov/36.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

►Here $k=|𝐤|$, and $𝐕$ is the ship velocity (so that $\mathrm{V}=|𝐕|$). …

6: 9.1 Special Notation

… ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

7: 13.28 Physical Applications

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$f_1(\xi )={\xi }^{-\frac{1}{2}}{V}_{\kappa ,\frac{1}{2}p}^{(1)}(2\mathrm{i}k\xi )$ ,

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$f_2(\eta )={\eta }^{-\frac{1}{2}}{V}_{\kappa ,\frac{1}{2}p}^{(2)}(-2\mathrm{i}k\eta )$ ,

►and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …

8: 19.33 Triaxial Ellipsoids

… ►The surface area of an ellipsoid with semiaxes $a,b,c$, and volume $V=4\pi abc/3$ is given by ►

19.33.1 $$S=3V{R}_G(a^{-2},b^{-2},c^{-2}),$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $V$: volume and $S$: area

Referenced by:

§19.33(i)

Permalink:

http://dlmf.nist.gov/19.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(i), §19.33 and Ch.19

… ►The potential is …and the electric capacity $C=Q/V(0)$ is given by … ►Let a homogeneous magnetic ellipsoid with semiaxes $a,b,c$, volume $V=4\pi abc/3$, and susceptibility $\chi$ be placed in a previously uniform magnetic field $H$ parallel to the principal axis with semiaxis $c$. …

9: 28.32 Mathematical Applications

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28.32.2 $$\frac{{\partial }^{2}V}{{\partial x}^2}+\frac{{\partial }^{2}V}{{\partial y}^2}+k^{2}V=0$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

►then becomes ►

28.32.3 $$\frac{{\partial }^{2}V}{{\partial \xi }^2}+\frac{{\partial }^{2}V}{{\partial \eta }^2}+\frac{1}{2}{c}^2{k}^2(\cosh \left(2\xi \right)-\cos \left(2\eta \right))V=0.$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\eta$: variable, $\xi$: variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.32(i), §28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

►The separated solutions $V(\xi ,\eta )=v(\xi )w(\eta )$ can be obtained from the modified Mathieu’s equation (28.20.1) for $v$ and from Mathieu’s equation (28.2.1) for $w$, where $a$ is the separation constant and $q=\frac{1}{4}{c}^2{k}^2$. … ►

28.32.8 $${\nabla }^{2}V+k^{2}V=0$$

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Symbols:

$k$: parameter

Permalink:

http://dlmf.nist.gov/28.32.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(ii), §28.32 and Ch.28

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10: 12.2 Differential Equations

… ►Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where … ►For real values of $z$ $(=x)$, numerically satisfactory pairs of solutions (§2.7(iv)) of (12.2.2) are $U(a,x)$ and $V(a,x)$ when $x$ is positive, or $U(a,-x)$ and $V(a,-x)$ when $x$ is negative. … ►

12.2.10 $$𝒲\left\{U(a,z),V(a,z)\right\}=\sqrt{2/\pi },$$

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Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.4.1

Permalink:

http://dlmf.nist.gov/12.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(iii), §12.2 and Ch.12

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12.2.14 $$V(n+\frac{1}{2},-z)={(-1)}^{n}V(n+\frac{1}{2},z).$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/12.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(iv), §12.2 and Ch.12

… ►Properties of $\overline{U}(a,x)$ follow immediately from those of $V(a,x)$ via (12.2.21). …