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11: 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 of $f$ with respect to $x$ , $z$: 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 of $f$ with respect to $x$ , $z$: 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 of $f$ with respect to $x$ , $z$: 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 of $f$ with respect to $x$ , $z$: 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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12: 12.17 Physical Applications

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

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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 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

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

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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

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$$\frac{d^{2}U}{{d\xi }^2}+\left(\sigma {\xi }^2+\lambda \right)U=0,$$

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$$\frac{d^{2}V}{{d\eta }^2}+\left(\sigma {\eta }^2-\lambda \right)V=0,$$

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$$\frac{d^{2}W}{{d\zeta }^2}+\left(k^2-\sigma \right)W=0,$$

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13: 12.15 Generalized Parabolic Cylinder Functions

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12.15.1 $$\frac{d^{2}w}{{dz}^2}+\left(\nu +{\lambda }^{-1}-{\lambda }^{-2}{z}^{\lambda }\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex parameter

Referenced by:

§12.15

Permalink:

http://dlmf.nist.gov/12.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.15 and Ch.12

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14: 10.38 Derivatives with Respect to Order

§10.38 Derivatives with Respect to Order

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10.38.2 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }=\frac{1}{2}\pi \csc \left(\nu \pi \right)\left(\frac{\partial I_{-\nu }\left(z\right)}{\partial \nu }-\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }\right)-\pi \cot \left(\nu \pi \right)K_{\nu }\left(z\right),$$ $\nu \notin \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\mathbb{Z}$: set of all integers, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.43

Referenced by:

(10.38.1), §10.38, §10.38

Permalink:

http://dlmf.nist.gov/10.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38 and Ch.10

… ►For $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4). … ►

${{\displaystyle \frac{\partial I_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=-K_0\left(z\right),$

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${{\displaystyle \frac{\partial K_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=0.$

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15: 19.4 Derivatives and Differential Equations

§19.4 Derivatives and Differential Equations

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§19.4(i) Derivatives

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${\displaystyle \frac{dE\left(k\right)}{dk}}={\displaystyle \frac{E\left(k\right)-K\left(k\right)}{k}},$

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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 of $f$ with respect to $x$ , $k$: 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

… ►Let $D_k=\partial /\partial k$. …

16: 1.13 Differential Equations

… ►A solution becomes unique, for example, when $w$ and $dw/dz$ are prescribed at a point in $D$. … ►

Elimination of First Derivative by Change of Dependent Variable

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Elimination of First Derivative by Change of Independent Variable

… ►Here dots denote differentiations with respect to $\zeta$, and $\{z,\zeta \}$ is the Schwarzian derivative: … ►

Cayley’s Identity

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17: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

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§13.3(ii) Differentiation Formulas

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13.3.22 $$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.4.21

Permalink:

http://dlmf.nist.gov/13.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

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13.3.23 $$\frac{d^n}{{dz}^n}U(a,b,z)={(-1)}^n{\left(a\right)}_{n}U(a+n,b+n,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

A&S Ref:

13.4.22

Referenced by:

§13.3(i)

Permalink:

http://dlmf.nist.gov/13.3.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

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13.3.29 $${\left(z\frac{d}{dz}z\right)}^n=z^n\frac{d^n}{{dz}^n}{z}^n,$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.15(ii), §13.3(ii)

Permalink:

http://dlmf.nist.gov/13.3.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

18: 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 of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: 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 of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: 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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19: 12.8 Recurrence Relations and Derivatives

§12.8 Recurrence Relations and Derivatives

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§12.8(ii) Derivatives

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12.8.9 $$\frac{d^m}{{dz}^m}\left({\mathrm{e}}^{\frac{1}{4}{z}^2}U(a,z)\right)={(-1)}^m{\left(\frac{1}{2}+a\right)}_m{\mathrm{e}}^{\frac{1}{4}{z}^2}U(a+m,z),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§12.8(ii)

Permalink:

http://dlmf.nist.gov/12.8.E9

Encodings:

pMML, png, TeX

See also:

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

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12.8.10 $$\frac{d^m}{{dz}^m}\left({\mathrm{e}}^{-\frac{1}{4}{z}^2}U(a,z)\right)={(-1)}^m{\mathrm{e}}^{-\frac{1}{4}{z}^2}U(a-m,z),$$

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Symbols:

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

Referenced by:

§12.8(ii)

Permalink:

http://dlmf.nist.gov/12.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(ii), §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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20: 32.2 Differential Equations

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32.2.1 $$\frac{d^{2}w}{{dz}^2}=6w^2+z,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: real

Referenced by:

§32.11(i), §32.11(i)

Permalink:

http://dlmf.nist.gov/32.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(i), §32.2 and Ch.32

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32.2.2 $$\frac{d^{2}w}{{dz}^2}=2w^3+zw+\alpha ,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: real and $\alpha$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(i), §32.2 and Ch.32

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32.2.7 $$\frac{d^{2}w}{{dz}^2}=F(z,w,\frac{dw}{dz}),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: real

Permalink:

http://dlmf.nist.gov/32.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(i), §32.2 and Ch.32

►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►

32.2.14 $$𝐼=z(1-z)\frac{d^2}{{dz}^2}+(1-2z)\frac{d}{dz}-\frac{1}{4}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: real and $𝐼$: operator

Permalink:

http://dlmf.nist.gov/32.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(iv), §32.2 and Ch.32

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