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11: 19.18 Derivatives and Differential Equations

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§19.18(ii) Differential Equations

… ►and two similar equations obtained by permuting $x,y,z$ in (19.18.10). … ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). ►The function $w=R_{-a}(\frac{1}{2},\frac{1}{2};x+y,x-y)$ satisfies an Euler–Poisson–Darboux equation: …Similarly, the function $u=R_{-a}(\frac{1}{2},\frac{1}{2};x+\mathrm{i}y,x-\mathrm{i}y)$ satisfies an equation of axially symmetric potential theory: …

12: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …

13: Bibliography E

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Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.

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External Links:

Document, MathReview (E. Rothe), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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14: 28.32 Mathematical Applications

… ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ►defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $ℂ$. …

15: Bibliography S

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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.

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External Links:

ISSN 0036-1410, Document, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§28.32(i).

Encodings:

BibTeX

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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External Links:

Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

Encodings:

BibTeX

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Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

Encodings:

BibTeX

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16: 19.4 Derivatives and Differential Equations

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§19.4(ii) Differential Equations

… ►If $\varphi =\pi /2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. An analogous differential equation of third order for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is given in Byrd and Friedman (1971, 118.03).

17: Bibliography K

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Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.

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External Links:

ISSN 1073-2772, FullText, MathReview, ZentralBlatt

Referenced by:

§31.10(ii).

Encodings:

BibTeX

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Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.

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External Links:

ISSN 0893-4983, MathReview, ZentralBlatt

Referenced by:

§32.8(v).

Encodings:

BibTeX

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18: 28.28 Integrals, Integral Representations, and Integral Equations

§28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

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28.28.2 $$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: 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, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

A&S Ref:

20.7.34 20.7.35 20.7.36 20.7.37

Permalink:

http://dlmf.nist.gov/28.28.E2

Encodings:

pMML, png, TeX

See also:

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

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

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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, ${\mathrm{Mc}}_n^{(j)}(z,h)$: 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

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

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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

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19: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. …

20: Bibliography F

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Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.

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External Links:

ISSN 0378-4371, Document, MathReview

Referenced by:

6th item.

Encodings:

BibTeX

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