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11: 9.2 Differential Equation

§9.2 Differential Equation

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§9.2(i) Airy’s Equation

… ►

§9.2(ii) Initial Values

… ►

§9.2(iii) Numerically Satisfactory Pairs of Solutions

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§9.2(vi) Riccati Form of Differential Equation

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12: 31.2 Differential Equations

§31.2 Differential Equations

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§31.2(i) Heun’s Equation

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31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

ⓘ

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, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(i), §31.2 and Ch.31

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§31.2(v) Heun’s Equation Automorphisms

… ►

Composite Transformations

…

13: 28.8 Asymptotic Expansions for Large $q$

… ►

28.8.5 $$V_m(\xi )\sim \frac{1}{2^{4}h}\left(-D_{m+2}\left(\xi \right)-m(m-1)D_{m-2}\left(\xi \right)\right)+\frac{1}{2^{10}{h}^2}\left(D_{m+6}\left(\xi \right)+(m^2-25m-36)D_{m+2}\left(\xi \right)-m(m-1)(m^2+27m-10)D_{m-2}\left(\xi \right)-6!\left(\genfrac{}{}{0.0pt}{}{m}{6}\right)D_{m-6}\left(\xi \right)\right)+\mathrm{\cdots },$$

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Defines:

$V_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.18 (in slightly different form)

Referenced by:

Erratum (V1.0.16) for Equation (28.8.5)

Permalink:

http://dlmf.nist.gov/28.8.E5

Encodings:

pMML, png, TeX

Error (effective with 1.0.16):

Originally the $-$ in front of the $6!$ was given incorrectly as $+$.

Suggested 2017-02-02 by Daniel Karlsson

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

…

14: 7.12 Asymptotic Expansions

… ►

7.12.4 $$\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}}+R_n^{(\mathrm{f})}(z),$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

A&S Ref:

7.3.27 (in different form)

Referenced by:

§7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E4

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{f})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

►

7.12.5 $$\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi z^2/2)}^{2m+1}},+R_n^{(\mathrm{g})}(z),$$

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{g})}(z)$: remainder term

A&S Ref:

7.3.28 (in different form)

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E5

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{{\pi }^2{z}^3}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m+1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{g})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

…

15: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.7 $$\frac{1}{\pi }{\int }_{{ℒ}_j}{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.18 (in slightly different notation)

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

►

28.28.8 $$\frac{1}{\pi }{\int }_{{ℒ}_j}2\mathrm{i}h\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }^{\prime }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.19 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

… ►

28.28.21 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\cos \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Mc}}_{2m+1}^{(j)}(z,h),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.26 (in different form and only for $\mathrm{\ell }=0$)

Referenced by:

Erratum (V1.0.11) for Equations (28.28.21) and (28.28.22)

Permalink:

http://dlmf.nist.gov/28.28.E21

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the prefactor and uppper limit of integration were given incorrectly as ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$.

Reported 2015-05-20 by Ruslan Kabasayev

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

►

28.28.22 $$\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{se}}_{2m+1}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.27 (in different form and only for $\mathrm{\ell }=0$)

Referenced by:

Erratum (V1.0.11) for Equations (28.28.21) and (28.28.22)

Permalink:

http://dlmf.nist.gov/28.28.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the prefactor and uppper limit of integration were given incorrectly as ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$.

Reported 2015-05-20 by Ruslan Kabasayev

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

…

16: 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►

§30.13(iv) Separation of Variables

►The wave equation …Equations (30.13.9) and (30.13.10) agree with (30.2.1). … ►

17: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►

8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta ),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Referenced by:

Erratum (V1.0.16) for Equation (8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E5

Encodings:

pMML, png, TeX

Change (effective with 1.0.16):

To be consistent with the notation used in (8.12.16), the original equation,

$$\mathrm{\Gamma }\left(a+1\right)\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\pi \mathrm{i}}\mathrm{\Gamma }(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\mp \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)+\mathrm{i}T(a,\eta ),$$

was rewritten.

See also:

Annotations for §8.12 and Ch.8

… ►The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta =0$, and the Maclaurin series expansion of $c_k(\eta )$ is given by … ►A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by … ►

Inverse Function

►For asymptotic expansions, as $a\to \mathrm{\infty }$, of the inverse function $x=x(a,q)$ that satisfies the equation …

18: 20.13 Physical Applications

… ►with $\kappa =-\mathrm{i}\pi /4$. ►For $\tau =\mathrm{i}t$, with $\alpha ,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation …These two apparently different solutions differ only in their normalization and boundary conditions. …Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). … ►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

19: 29.12 Definitions

… ►Throughout §§29.12–29.16 the order $\nu$ in the differential equation (29.2.1) is assumed to be a nonnegative integer. … ►

29.12.1 $${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)={\mathrm{𝐸𝑐}}_{2n}^{2m}(z,k^2),$$

ⓘ

Defines:

${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$: Lamé polynomial

Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter

Referenced by:

§29.12(i)

Permalink:

http://dlmf.nist.gov/29.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(i), §29.12 and Ch.29

►

29.12.2 $${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)={\mathrm{𝐸𝑐}}_{2n+1}^{2m+1}(z,k^2),$$

ⓘ

Defines:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial

Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(i), §29.12 and Ch.29

►

29.12.3 $${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)={\mathrm{𝐸𝑠}}_{2n+1}^{2m+1}(z,k^2),$$

ⓘ

Defines:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial

Symbols:

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $m$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(i), §29.12 and Ch.29

… ►In the fourth column the variable $z$ and modulus $k$ of the Jacobian elliptic functions have been suppressed, and $P({\mathrm{sn}}^2)$ denotes a polynomial of degree $n$ in ${\mathrm{sn}}^2(z,k)$ (different for each type). …

20: 18.36 Miscellaneous Polynomials

… ►These are polynomials in one variable that are orthogonal with respect to a number of different measures. … ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … ►

18.36.2 $$L_n^{(-k)}\left(x\right)={(-1)}^k\frac{(n-k)!}{n!}{x}^k{L}_{n-k}^{(k)}\left(x\right),$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.36.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(v), §18.36 and Ch.18

… ►

18.36.7 $$T_k(y)\equiv -x{y}^{\prime \prime }+\frac{x-k}{x+k}((x+k+1)y^{\prime }-y)=(n-1)y.$$

ⓘ

Symbols:

$\equiv$: equals by definition, $y$: real variable, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.36(vi)

Permalink:

http://dlmf.nist.gov/18.36.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

… ►In §18.39(i) it is seen that the functions, $\sqrt{w(x)}{\widehat{H}}_{n+3}\left(x\right)$, are solutions of a Schrödinger equation with a rational potential energy; and, in spite of first appearances, the Sturm oscillation theorem, Simon (2005c, Theorem 3.3, p. 35), is satisfied. …