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21: 32.5 Integral Equations

… ►Let $K(z,\zeta )$ be the solution of ►

32.5.1 $$K(z,\zeta )=k\mathrm{Ai}\left(\frac{z+\zeta }{2}\right)+\frac{k^2}{4}{\int }_z^{\mathrm{\infty }}{\int }_z^{\mathrm{\infty }}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)\mathrm{Ai}\left(\frac{t+\zeta }{2}\right)dsdt,$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $z$: real, $k$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E1

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Annotations for §32.5 and Ch.32

… ►

32.5.2 $$w(z)=K(z,z),$$

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Symbols:

$z$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E2

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Annotations for §32.5 and Ch.32

…

22: 28.2 Definitions and Basic Properties

… ►

§28.2(iv) Floquet Solutions

►A solution with the pseudoperiodic property (28.2.14) is called a Floquet solution with respect to $\nu$. … ►The Fourier series of a Floquet solution …leads to a Floquet solution. … ►For the connection with the basic solutions in §28.2(ii), …

23: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …

24: 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

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

►

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

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

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

… ►

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

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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

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

…

25: 4.34 Derivatives and Differential Equations

… ►With $a\ne 0$, the general solutions of the differential equations ►

4.34.7 $$\frac{d^{2}w}{{dz}^2}-a^{2}w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $a$: real or complex constant, $z$: complex variable and $w$: solution

Permalink:

http://dlmf.nist.gov/4.34.E7

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Annotations for §4.34 and Ch.4

… ►

4.34.12 $$w=(1/a)\sinh \left(az+c\right),$$

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Symbols:

$\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E12

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Annotations for §4.34 and Ch.4

►

4.34.13 $$w=(1/a)\cosh \left(az+c\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E13

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Annotations for §4.34 and Ch.4

►

4.34.14 $$w=(1/a)\coth \left(az+c\right),$$

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Symbols:

$\coth z$: hyperbolic cotangent function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E14

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Annotations for §4.34 and Ch.4

…

26: 10.72 Mathematical Applications

… ►

§10.72(i) Differential Equations with Turning Points

►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …

27: Bibliography O

… ►

Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

Encodings:

BibTeX

… ►

On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

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External Links:

ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt

Referenced by:

§16.25, §2.7(ii), §3.7(iii).

Encodings:

BibTeX

… ►

Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §2.11(v), §9.7(v).

Encodings:

BibTeX

… ►

Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

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External Links:

ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17, §32.17.

Encodings:

BibTeX

… ►

Solution of Equations in Euclidean and Banach Spaces. Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London.

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Notes:

Third edition of Solution of equations and systems of equations

External Links:

MathReview (J. Burlak), ZentralBlatt

Referenced by:

§3.3(v), §3.8(iii).

Encodings:

BibTeX

…

28: 2.7 Differential Equations

… ►But there is an independent solution … ►Formal solutions are … ►has twice-continuously differentiable solutions … ►

§2.7(iv) Numerically Satisfactory Solutions

►One pair of independent solutions of the equation …

29: 32.10 Special Function Solutions

§32.10 Special Function Solutions

… ►with solution … ►

§32.10(iv) Fourth Painlevé Equation

… ►which has the solution … ►

30: 28.32 Mathematical Applications

… ►then becomes … ►Let $u(\zeta )$ be a solution of Mathieu’s equation (28.2.1) and $K(z,\zeta )$ be a solution of …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 $ℂ$. ►Kernels $K$ can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. … ►The first is the $2\pi$-periodicity of the solutions; the second can be their asymptotic form. …