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11: 16.14 Partial Differential Equations

§16.14 Partial Differential Equations

►

§16.14(i) Appell Functions

►

$$x(1-x)\frac{{\partial }^2{F}_1}{{\partial x}^2}+y(1-x)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_1}{\partial x}-\beta y\frac{\partial F_1}{\partial y}-\alpha \beta F_1=0,$$

►

$$y(1-y)\frac{{\partial }^2{F}_1}{{\partial y}^2}+x(1-y)\frac{{\partial }^2{F}_1}{\partial x\partial y}+\left(\gamma -(\alpha +{\beta }^{\prime }+1)y\right)\frac{\partial F_1}{\partial y}-{\beta }^{\prime }x\frac{\partial F_1}{\partial x}-\alpha {\beta }^{\prime }{F}_1=0,$$

… ►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …

12: 14.30 Spherical and Spheroidal Harmonics

… ►

14.30.10 $$\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial W}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial W}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}W}{{\partial \varphi }^2}=0,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function and $W(\rho ,\theta ,\varphi )$: solution

Permalink:

http://dlmf.nist.gov/14.30.E10

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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

…

13: 1.9 Calculus of a Complex Variable

… ►

§1.9(ii) Continuity, Point Sets, and Differentiation

… ►

Differentiation

… ►Differentiability automatically implies continuity. ►

Cauchy–Riemann Equations

… ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. …

14: 6.16 Mathematical Applications

… ►The $n$th partial sum is given by ►

6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

ⓘ

Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

15: 12.17 Physical Applications

… ►

12.17.2 $${\nabla }^2=\frac{{\partial }^2}{{\partial x}^2}+\frac{{\partial }^2}{{\partial y}^2}+\frac{{\partial }^2}{{\partial z}^2}$$

ⓘ

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 and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

… ►

12.17.4 $$\frac{1}{{\xi }^2+{\eta }^2}\left(\frac{{\partial }^{2}w}{{\partial \xi }^2}+\frac{{\partial }^{2}w}{{\partial \eta }^2}\right)+\frac{{\partial }^{2}w}{{\partial \zeta }^2}+k^{2}w=0.$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: constant, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate

Permalink:

http://dlmf.nist.gov/12.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.17 and Ch.12

…

16: 9.18 Tables

… ►

Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

… ►

Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

►

Zhang and Jin (1996, p. 339) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$; 8D.

… ►

Smirnov (1960) tabulates $U_1(x,\alpha )$, $U_2(x,\alpha )$, defined by (9.13.20), (9.13.21), and also $\partial U_1(x,\alpha )/\partial x$, $\partial U_2(x,\alpha )/\partial x$, for $\alpha =1$, $x=-6(.01)10$ to 5D or 5S, and also for $\alpha =\pm \frac{1}{4}$, $\pm \frac{1}{3}$, $\pm \frac{1}{2}$, $\pm \frac{2}{3}$, $\pm \frac{3}{4}$, $\frac{5}{4}$, $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, 2, $x=0(.01)6$; 4D.

17: 23.21 Physical Applications

… ►

§23.21(ii) Nonlinear Evolution Equations

►Airault et al. (1977) applies the function $\mathrm{\wp }$ to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … ►

23.21.2 $$(\eta -\zeta )(\zeta -\xi )(\xi -\eta ){\nabla }^2=(\zeta -\eta )f(\xi )f^{\prime }(\xi )\frac{\partial }{\partial \xi }+(\xi -\zeta )f(\eta )f^{\prime }(\eta )\frac{\partial }{\partial \eta }+(\eta -\xi )f(\zeta )f^{\prime }(\zeta )\frac{\partial }{\partial \zeta },$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\xi$: ellipsoidal coordinate, $\eta$: ellipsoidal coordinate, $\zeta$: ellipsoidal coordinate and $f(\rho )$: function

Permalink:

http://dlmf.nist.gov/23.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

… ►

23.21.5 $$\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(w\right)\right)\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(u\right)\right)\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(v\right)\right){\nabla }^2=\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(v\right)\right)\frac{{\partial }^2}{{\partial u}^2}+\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(w\right)\right)\frac{{\partial }^2}{{\partial v}^2}+\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(u\right)\right)\frac{{\partial }^2}{{\partial w}^2}.$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝕃$: lattice, $u$: change of variable, $v$: change of variable and $w$: change of variable

Permalink:

http://dlmf.nist.gov/23.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

…

18: 3.8 Nonlinear Equations

… ►

3.8.13 $$\frac{dz}{d\alpha }=-\frac{\partial f}{\partial \alpha }/\frac{\partial f}{\partial z}.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/3.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8 and Ch.3

… ►

3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

ⓘ

Permalink:

http://dlmf.nist.gov/3.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. … ►

3.8.16 $$\frac{dx}{d{a}_{19}}=-\frac{{20}^{19}}{19!}=(-4.30\mathrm{\dots })\times {10}^7.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/3.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

…

19: 28.32 Mathematical Applications

… ►

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

ⓘ

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

… ►

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

ⓘ

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

… ►

28.32.4 $$\frac{{\partial }^{2}K}{{\partial z}^2}-\frac{{\partial }^{2}K}{{\partial \zeta }^2}=2q\left(\cos \left(2z\right)-\cos \left(2\zeta \right)\right)K.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $q=h^2$: parameter, $z$: complex variable, $\zeta$: variable and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

28.32.5 $$K(z,\zeta )\frac{du(\zeta )}{d\zeta }-u(\zeta )\frac{\partial K(z,\zeta )}{\partial \zeta }$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $\zeta$: variable, $u(\zeta )$: solution and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E5

Encodings:

pMML, png, TeX

See also:

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

…

20: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation ►

20.13.1 $$\partial \theta (z|\tau )/\partial \tau =\kappa {\partial }^2\theta (z|\tau )/{\partial z}^2,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\tau$: lattice parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

… ►

20.13.2 $$\partial \theta /\partial t=\alpha {\partial }^2\theta /{\partial z}^2,$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: time, $\alpha$: real parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

…