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11: 16.14 Partial Differential Equations

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

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

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$$x(1-x)\frac{{\partial }^2{F}_2}{{\partial x}^2}-xy\frac{{\partial }^2{F}_2}{\partial x\partial y}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_2}{\partial x}-\beta y\frac{\partial F_2}{\partial y}-\alpha \beta F_2=0,$$

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$$x(1-x)\frac{{\partial }^2{F}_4}{{\partial x}^2}-2xy\frac{{\partial }^2{F}_4}{\partial x\partial y}-y^2\frac{{\partial }^2{F}_4}{{\partial y}^2}+\left(\gamma -(\alpha +\beta +1)x\right)\frac{\partial F_4}{\partial x}-(\alpha +\beta +1)y\frac{\partial F_4}{\partial y}-\alpha \beta F_4=0,$$

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$$y(1-y)\frac{{\partial }^2{F}_4}{{\partial y}^2}-2xy\frac{{\partial }^2{F}_4}{\partial x\partial y}-x^2\frac{{\partial }^2{F}_4}{{\partial x}^2}+\left({\gamma }^{\prime }-(\alpha +\beta +1)y\right)\frac{\partial F_4}{\partial y}-(\alpha +\beta +1)x\frac{\partial F_4}{\partial x}-\alpha \beta F_4=0.$$

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12: 10.15 Derivatives with Respect to Order

§10.15 Derivatives with Respect to Order

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10.15.2 $$\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }=\cot \left(\nu \pi \right)\left(\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }-\pi Y_{\nu }\left(z\right)\right)-\csc \left(\nu \pi \right)\frac{\partial J_{-\nu }\left(z\right)}{\partial \nu }-\pi J_{\nu }\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.65

Referenced by:

(10.15.1), §10.15

Permalink:

http://dlmf.nist.gov/10.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

… ►For $\partial J_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.2.4) and (10.15.3). … ►

10.15.5 $${\frac{\partial J_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=\frac{\pi }{2}{Y}_0\left(z\right),{\frac{\partial Y_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =0}=-\frac{\pi }{2}{J}_0\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.68

Referenced by:

§10.15, §10.38

Permalink:

http://dlmf.nist.gov/10.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.15, §10.15 and Ch.10

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13: 19.4 Derivatives and Differential Equations

§19.4 Derivatives and Differential Equations

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§19.4(i) Derivatives

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19.4.3 $$\frac{d^{2}E\left(k\right)}{{dk}^2}=-\frac{1}{k}\frac{dK\left(k\right)}{dk}=\frac{k^{\prime }{}^{2}K\left(k\right)-E\left(k\right)}{k^{2}k^{\prime }{}^2},$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$: derivative, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.6 $$\frac{\partial E(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-F(\varphi ,k)}{k},$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

… ►Let $D_k=\partial /\partial k$. …

14: 10.73 Physical Applications

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10.73.1 $${\nabla }^{2}V=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{{\partial \varphi }^2}+\frac{{\partial }^{2}V}{{\partial z}^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/10.73.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

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10.73.2 $${\nabla }^2\psi =\frac{1}{c^2}\frac{{\partial }^2\psi }{{\partial t}^2},$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/10.73.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

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10.73.3 $${\nabla }^{4}W+{\lambda }^2\frac{{\partial }^{2}W}{{\partial t}^2}=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative and $\partial x$: partial differential

Permalink:

http://dlmf.nist.gov/10.73.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(i), §10.73 and Ch.10

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10.73.4 $$({\nabla }^2+k^2)f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+k^{2}f.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\sin z$: sine function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.73.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.73(ii), §10.73 and Ch.10

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15: 1.13 Differential Equations

… ►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ►Then at each $z\in D$, $w$, $\partial w/\partial z$ and ${\partial }^{2}w/{\partial z}^2$ are analytic functions of $u$. … ►

Elimination of First Derivative by Change of Dependent Variable

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Elimination of First Derivative by Change of Independent Variable

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Cayley’s Identity

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16: 30.12 Generalized and Coulomb Spheroidal Functions

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30.12.1 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +\alpha z+{\gamma }^2(1-z^2)-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.12

Permalink:

http://dlmf.nist.gov/30.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

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30.12.2 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +{\gamma }^2(1-z^2)-\frac{\alpha (\alpha +1)}{z^2}-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Permalink:

http://dlmf.nist.gov/30.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

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17: 14.29 Generalizations

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14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

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18: 31.12 Confluent Forms of Heun’s Equation

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31.12.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+ϵ\right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.2 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\delta }{z^2}+\frac{\gamma }{z}+1\right)\frac{dw}{dz}+\frac{\alpha z-q}{z^2}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

Erratum (V1.0.7) for Equation (31.12.3)

Permalink:

http://dlmf.nist.gov/31.12.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.4 $$\frac{d^{2}w}{{dz}^2}+\left(\gamma +z\right)z\frac{dw}{dz}+\left(\alpha z-q\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative, $z$: complex variable, $\gamma$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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19: 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, $\partial x$: partial differential, $x$: 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

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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, $\partial x$: partial differential, $\eta$: 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 $$\frac{{\partial }^{2}K}{{\partial z}^2}-\frac{{\partial }^{2}K}{{\partial \zeta }^2}=2q\left(\cos \left(2z\right)-\cos \left(2\zeta \right)\right)K.$$

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Symbols:

$\cos z$: cosine function, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $q=h^2$: 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

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

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Symbols:

$\frac{df}{dx}$: derivative, $\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: 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: 30.13 Wave Equation in Prolate Spheroidal Coordinates

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30.13.3 $$h_{\xi }^2={\left(\frac{\partial x}{\partial \xi }\right)}^2+{\left(\frac{\partial y}{\partial \xi }\right)}^2+{\left(\frac{\partial z}{\partial \xi }\right)}^2=\frac{c^2({\xi }^2-{\eta }^2)}{{\xi }^2-1},$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

A&S Ref:

21.4.2

Permalink:

http://dlmf.nist.gov/30.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

►

30.13.4 $$h_{\eta }^2={\left(\frac{\partial x}{\partial \eta }\right)}^2+{\left(\frac{\partial y}{\partial \eta }\right)}^2+{\left(\frac{\partial z}{\partial \eta }\right)}^2=\frac{c^2({\xi }^2-{\eta }^2)}{1-{\eta }^2},$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

Permalink:

http://dlmf.nist.gov/30.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

►

30.13.5 $$h_{\varphi }^2={\left(\frac{\partial x}{\partial \varphi }\right)}^2+{\left(\frac{\partial y}{\partial \varphi }\right)}^2+{\left(\frac{\partial z}{\partial \varphi }\right)}^2=c^2({\xi }^2-1)(1-{\eta }^2).$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $z$: complex variable, $x$: real variable, $y$: real variable, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $\varphi$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

Permalink:

http://dlmf.nist.gov/30.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(ii), §30.13 and Ch.30

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30.13.6 $${\nabla }^2=\frac{1}{h_{\xi }{h}_{\eta }{h}_{\varphi }}\left(\frac{\partial }{\partial \xi }\left(\frac{h_{\eta }{h}_{\varphi }}{h_{\xi }}\frac{\partial }{\partial \xi }\right)+\frac{\partial }{\partial \eta }\left(\frac{h_{\xi }{h}_{\varphi }}{h_{\eta }}\frac{\partial }{\partial \eta }\right)+\frac{\partial }{\partial \varphi }\left(\frac{h_{\xi }{h}_{\eta }}{h_{\varphi }}\frac{\partial }{\partial \varphi }\right)\right)=\frac{1}{c^2({\xi }^2-{\eta }^2)}\left(\frac{\partial }{\partial \xi }\left(({\xi }^2-1)\frac{\partial }{\partial \xi }\right)+\frac{\partial }{\partial \eta }\left((1-{\eta }^2)\frac{\partial }{\partial \eta }\right)+\frac{{\xi }^2-{\eta }^2}{({\xi }^2-1)(1-{\eta }^2)}\frac{{\partial }^2}{{\partial \varphi }^2}\right).$$

ⓘ

Symbols:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $\varphi$: prolate spheroidal coordinate, $c$: positive constant and $h_{\xi },h_{\eta },h_{\varphi }$: metric coefficients

A&S Ref:

21.5.1 (in corrected form)

Permalink:

http://dlmf.nist.gov/30.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iii), §30.13 and Ch.30

… ►

30.13.11 $$\frac{d^2{w}_3}{{d\varphi }^2}+{\mu }^2{w}_3=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative, $\varphi$: prolate spheroidal coordinate, $w_j$: solution to DE and $\mu$: real parameter

Referenced by:

§30.14(iv)

Permalink:

http://dlmf.nist.gov/30.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

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