# elementary solutions

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##### 1: 32.9 Other Elementary Solutions

###### §32.9 Other Elementary Solutions

… ►Elementary nonrational solutions of ${\text{P}}_{\text{III}}$ are … ►Elementary nonrational solutions of ${\text{P}}_{\text{V}}$ are … ►An elementary algebraic solution of ${\text{P}}_{\text{VI}}$ is … ►##### 2: 32.8 Rational Solutions

###### §32.8 Rational Solutions

…##### 3: Bibliography F

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On a unified approach to transformations and elementary solutions of Painlevé equations.
J. Math. Phys. 23 (11), pp. 2033–2042.
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##### 4: Bibliography L

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Elementary solutions of certain Painlevé equations.
Differ. Uravn. 1 (3), pp. 731–735 (Russian).
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##### 5: 28.8 Asymptotic Expansions for Large $q$

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►With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii).
►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions
${\mathrm{me}}_{\nu}(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu}^{(j)}(z,h)$ (§28.20(iii)).
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##### 6: 32.2 Differential Equations

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►For arbitrary values of the parameters $\alpha $, $\beta $, $\gamma $, and $\delta $, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are

*transcendental*, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …##### 7: 2.8 Differential Equations with a Parameter

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►Corresponding to each positive integer $n$ there are solutions
${W}_{n,j}(u,\xi )$, $j=1,2$, that depend on arbitrarily chosen reference points ${\alpha}_{j}$, are ${C}^{\mathrm{\infty}}$ or analytic on $\mathbf{\Delta}$, and as $u\to \mathrm{\infty}$
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##### 8: 12.10 Uniform Asymptotic Expansions for Large Parameter

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12.10.30
$${\overline{v}}_{s}(t)={\mathrm{i}}^{s}{v}_{s}(-\mathrm{i}t).$$

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##### 9: 14.31 Other Applications

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►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)).
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###### §14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …##### 10: 22.19 Physical Applications

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►The bounded $(-\pi \le \theta \le \pi )$ oscillatory solution of (22.19.1) is traditionally written
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►This formulation gives the bounded and unbounded solutions from the same formula (22.19.3), for $k\ge 1$ and $k\le 1$, respectively.
…Figure 22.19.1 shows the nature of the solutions
$\theta (t)$ of (22.19.3) by graphing $\mathrm{am}(x,k)$ for both $0\le k\le 1$, as in Figure 22.16.1, and $k\ge 1$, where it is periodic.
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►Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions.
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►Elementary discussions of this topic appear in Lawden (1989, §5.7), Greenhill (1959, pp. 101–103), and Whittaker (1964, Chapter VI).
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