periodic solutions

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11: 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

4K

. …

12: 29.8 Integral Equations

… ►Let

w(z)

be any solution of (29.2.1) of period

4K

w_{2}(z)

be a linearly independent solution, and

\mathscr{W}\left\{w,w_{2}\right\}

denote their Wronskian. …

13: 29.19 Physical Applications

… ►Simply-periodic Lamé functions (

\nu

noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. … ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. Shail (1978) treats applications to solutions of elliptic crack and punch problems. Hargrave (1978) studies high frequency solutions of the delta wing equation. …

14: 28.12 Definitions and Basic Properties

… ►When

\nu=s/m

is a rational number, but not an integer, all solutions of Mathieu’s equation are periodic with period

2m\pi

. …

15: 22.19 Physical Applications

… ►Figure 22.19.1 shows the nature of the solutions

\theta(t)

of (22.19.3) by graphing

\operatorname{am}\left(x,k\right)

for both

0\leq k\leq 1

, as in Figure 22.16.1, and

k\geq 1

, where it is periodic. … ►Such solutions include standing or stationary waves, periodic cnoidal waves, and single and multi-solitons occurring in diverse physical situations such as water waves, optical pulses, quantum fluids, and electrical impulses (Hasegawa (1989), Carr et al. (2000), Kivshar and Luther-Davies (1998), and Boyd (1998, Appendix D2.2)). …

16: 29.10 Lamé Functions with Imaginary Periods

§29.10 Lamé Functions with Imaginary Periods

… ►

29.10.3

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z^{\prime}}^{2}}+(h^{\prime}-\nu(\nu+1){k^% {\prime}}^{2}{\operatorname{sn}}^{2}\left(z^{\prime},k^{\prime}\right))w=0.

ⓘ

Symbols:: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $h$ : real parameter, $k$ : real parameter, $\nu$ : real parameter and $w(z)$ : solution
Permalink:: http://dlmf.nist.gov/29.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §29.10 and Ch.29

… ►are solutions of (29.2.1). The first and the fourth functions have period

2\mathrm{i}{K^{\prime}}

; the second and the third have period

4\mathrm{i}{K^{\prime}}

. …

17: 28.2 Definitions and Basic Properties

… ►

§28.2(vi) Eigenfunctions

…

18: 1.13 Differential Equations

… ►

Fundamental Pair

… ► … ►

§1.13(v) Products of Solutions

… ►

§1.13(vii) Closed-Form Solutions

… ►or periodic boundary conditions …

19: 29.17 Other Solutions

§29.17 Other Solutions

►

§29.17(i) Second Solution

►If (29.2.1) admits a Lamé polynomial solution

E

, then a second linearly independent solution

F

is given by …For properties of these solutions see Arscott (1964b, §9.7), Erdélyi et al. (1955, §15.5.1), Shail (1980), and Sleeman (1966b). … ►They are algebraic functions of

\operatorname{sn}\left(z,k\right)

\operatorname{cn}\left(z,k\right)

, and

\operatorname{dn}\left(z,k\right)

, and have primitive period

8K

. …

20: 32.10 Special Function Solutions

§32.10 Special Function Solutions

… ►with solution … ►which has the solution … ►where the fundamental periods

2\phi_{1}

and

2\phi_{2}

are linearly independent functions satisfying the hypergeometric equation … ►