difference equation

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31: 7.12 Asymptotic Expansions

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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

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

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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

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32: 1.16 Distributions

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1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

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Defines:

$𝐃$: vector differential 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 $x$ , $n$: nonnegative integer, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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

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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, $a$: 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

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Composite Transformations

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34: Bibliography K

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

BibTeX

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35: 3.3 Interpolation

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§3.3(iii) Divided Differences

… ►Explicitly, the divided difference of order $n$ is given by … ►This represents the Lagrange interpolation polynomial in terms of divided differences: …Newton’s formula has the advantage of allowing easy updating: incorporation of a new point $z_{n+1}$ requires only addition of the term with $\left[z_0,z_1,\mathrm{\dots },z_{n+1}\right]f$ to (3.3.38), plus the computation of this divided difference. …For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. …

36: 28.15 Expansions for Small $q$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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37: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

… ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi ,\eta ,\varphi )$, admits solutions of the form (30.13.8), where $w_1$ satisfies the differential equation …Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution $z=\pm \mathrm{i}\xi$. … ►

38: 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►The wave equation …where $w_1$, $w_2$, $w_3$ satisfy the differential equations …Equations (30.13.9) and (30.13.10) agree with (30.2.1). … ►

39: 27.5 Inversion Formulas

… ►For example, the equation $\zeta \left(s\right)\cdot (1/\zeta \left(s\right))=1$ is equivalent to the identity ►

27.5.2 $$\sum _{d|n}\mu \left(d\right)=\lfloor \frac{1}{n}\rfloor ,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\lfloor x\rfloor$: floor of $x$ , $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.1 II.B (in slightly different form)

Referenced by:

§27.5

Permalink:

http://dlmf.nist.gov/27.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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40: 28.10 Integral Equations

§28.10 Integral Equations

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§28.10(i) Equations with Elementary Kernels

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28.10.1 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos \left(2h\cos z\cos t\right){\mathrm{ce}}_{2n}(t,h^2)dt=\frac{A_0^{2n}(h^2)}{{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}{\mathrm{ce}}_{2n}(z,h^2),$$

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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, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.20 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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§28.10(ii) Equations with Bessel-Function Kernels

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§28.10(iii) Further Equations

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