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11: Bibliography D

… ►

The Radon Transform and Some of Its Applications. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:

Reprinted by Dover in 2007.

External Links:

ISBN 0-471-89804-X, MathReview (W. R. Madych), ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

… ►

Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:

Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.

External Links:

ISBN 0-8284-0225-6, Link, MathReview

Referenced by:

§27.11.

Encodings:

BibTeX

►

Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.32(iii).

Encodings:

BibTeX

… ►

Hydrodynamic Stability. Cambridge University Press, Cambridge.

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Notes:

Cambridge Monographs on Mechanics and Applied Mathematics

External Links:

ISBN 0-521-22798-4, MathReview (Adelina Georgescu), ZentralBlatt

Referenced by:

(9.13.25), (9.13.27), (9.13.28), (9.13.29), (9.13.30), (9.13.35), (9.13.36), (9.13.37), Figure 9.13.1, Figure 9.13.1, Figure 9.13.2, Figure 9.13.2, §9.13(ii), §9.13(ii), §9.16.

Encodings:

BibTeX

… ►

Uniform asymptotic approximations for the Whittaker functions $M_{\kappa ,i\mu }(z)$ and $W_{\kappa ,i\mu }(z)$ . Anal. Appl. (Singap.) 1 (2), pp. 199–212.

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External Links:

ISSN 0219-5305, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.20(v).

Encodings:

BibTeX

…

12: 4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). ►For an application of the Lambert $W$-function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). For other applications of the Lambert $W$-function see Corless et al. (1996).

13: 9.11 Products

… ►where $w_1$ and $w_2$ are any solutions of (9.2.1). For example, $w={\mathrm{Ai}}^2\left(z\right)$, $\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right)$, $\mathrm{Ai}\left(z\right)\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)$, $M^2\left(z\right)$. … ►Let $w_1,w_2$ be any solutions of (9.2.1), not necessarily distinct. … ►

9.11.10 $$\int z{w}_1^{\prime }{w}_2^{\prime }dz=\frac{3}{10}(-w_1{w}_2+z{w}_1{w}_2^{\prime }+z{w}_1^{\prime }{w}_2)+\frac{1}{5}(z^2{w}_1^{\prime }{w}_2^{\prime }-z^3{w}_1{w}_2).$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: function

Source:

Albright (1977, (A.21))

Permalink:

http://dlmf.nist.gov/9.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

►For $\int z^n{w}_1{w}_{2}dz$, $\int z^n{w}_1{w}_2^{\prime }dz$, $\int z^n{w}_1^{\prime }{w}_2^{\prime }dz$, where $n$ is any positive integer, see Albright (1977). …

14: 10.13 Other Differential Equations

… ►

10.13.4 $$w^{\prime \prime }+\frac{1\mp 2\nu }{z}{w}^{\prime }+{\lambda }^{2}w=0,$$ $w=z^{\pm \nu }{𝒞}_{\nu }\left(\lambda z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.52

Referenced by:

Erratum (V1.0.8) for Equation (10.13.4)

Permalink:

http://dlmf.nist.gov/10.13.E4

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.8):

This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu }{𝒞}_{\nu }\left(\lambda z\right)$ was covered.

Suggested 2014-01-27 by Tom Koornwinder

See also:

Annotations for §10.13 and Ch.10

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10.13.5 $$z^2{w}^{\prime \prime }+(1-2r)z{w}^{\prime }+({\lambda }^2{q}^2{z}^{2q}+r^2-{\nu }^2{q}^2)w=0,$$ $w=z^r{𝒞}_{\nu }\left(\lambda z^q\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.53

Permalink:

http://dlmf.nist.gov/10.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

… ►

10.13.7 $$z^2(z^2-{\nu }^2)w^{\prime \prime }+z(z^2-3{\nu }^2)w^{\prime }+({(z^2-{\nu }^2)}^2-(z^2+{\nu }^2))w=0,$$ $w={𝒞}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.55

Permalink:

http://dlmf.nist.gov/10.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.9 $$z^2{w}^{\prime \prime \prime }+3z{w}^{\prime \prime }+(4z^2+1-4{\nu }^2)w^{\prime }+4zw=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.58 (modified)

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

►

10.13.10 $$z^3{w}^{\prime \prime \prime }+z(4z^2+1-4{\nu }^2)w^{\prime }+(4{\nu }^2-1)w=0,$$ $w=z{𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.59

Permalink:

http://dlmf.nist.gov/10.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

…

15: 1.13 Differential Equations

… ►The equation … ►Two solutions $w_1(z)$ and $w_2(z)$ are called a fundamental pair if any other solution $w(z)$ is expressible as … ►The Wronskian of $w_1(z)$ and $w_2(z)$ is defined by … ►The following three statements are equivalent: $w_1(z)$ and $w_2(z)$ comprise a fundamental pair in $D$; $𝒲\left\{w_1(z),w_2(z)\right\}$ does not vanish in $D$; $w_1(z)$ and $w_2(z)$ are linearly independent, that is, the only constants $A$ and $B$ such that … ►then $W=UV$ is a solution of …

16: 10.36 Other Differential Equations

… ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

►

10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

…

17: 3.6 Linear Difference Equations

… ►Given numerical values of $w_0$ and $w_1$, the solution $w_n$ of the equation … ►But suppose that $w_n$ is a nontrivial solution such that … ►Then computation of $w_n$ by forward recursion is unstable. … ►beginning with $e_0=w_0$. …Then $w_n$ is generated by backward recursion from …

18: 28.30 Expansions in Series of Eigenfunctions

… ►Let ${\widehat{\lambda }}_m$, $m=0,1,2,\mathrm{\dots }$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_m(x)$, $m=0,1,2,\mathrm{\dots }$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus ►

28.30.1 $$w_m^{\prime \prime }+({\widehat{\lambda }}_m+Q(x))w_m=0,$$

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Symbols:

$m$: integer, $x$: real variable, $w(z)$: Mathieu’s equation solution and ${\widehat{\lambda }}_m$: characteristic values

Permalink:

http://dlmf.nist.gov/28.30.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

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28.30.2 $$\frac{1}{2\pi }{\int }_0^{2\pi }{w}_m(x)w_n(x)dx={\delta }_{m,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $m$: integer, $n$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

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28.30.3 $$f(x)=\sum _{m=0}^{\mathrm{\infty }}{f}_m{w}_m(x),$$

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Symbols:

$m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

… ►

28.30.4 $$f_m=\frac{1}{2\pi }{\int }_0^{2\pi }f(x)w_m(x)dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

…

19: 32.7 Bäcklund Transformations

… ►Let $w=w(z;\alpha )$ be a solution of ${\text{P}}_{\text{II}}$. … ►The solutions $w_{\alpha }=w(z;\alpha )$, $w_{\alpha \pm 1}=w(z;\alpha \pm 1)$, satisfy the nonlinear recurrence relation … ►Next, let $W_j=W(z;{\alpha }_j,{\beta }_j,1,-1)$, $j=0,1,2,3,4$, be solutions of ${\text{P}}_{\text{III}}$ with … ►Let $w=w(z;\alpha ,\beta ,1,-1)$ be a solution of ${\text{P}}_{\text{III}}$ and … ►Let $w=w(z;\alpha ,\beta ,\gamma ,\delta )$ be a solution of ${\text{P}}_{\text{VI}}$. …

20: 3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►By repeated differentiation of (3.7.1) all derivatives of $w(z)$ can be expressed in terms of $w(z)$ and $w^{\prime }(z)$ as follows. … ►Also let $𝐰$ denote the $(2P+2)\times 1$ vector … ►For $w^{\prime }=f(z,w)$ the standard fourth-order rule reads … ►For $w^{\prime \prime }=f(z,w,w^{\prime })$ the standard fourth-order rule reads …