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1: 22 Jacobian Elliptic Functions

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2: Vadim B. Kuznetsov

… ►He was also editor of The Kowalevski Property, CRM Proceedings and Lecture Notes 32, published by the American Mathematical Society in 2002. …

3: 14.29 Generalizations

… ►For properties see Virchenko and Fedotova (2001) and Braaksma and Meulenbeld (1967). … ►For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).

4: 32.16 Physical Applications

§32.16 Physical Applications

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5: Mark J. Ablowitz

… ►Their similarity solutions lead to special ODEs which have the Painlevé property; i. …ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents. …

6: 18.3 Definitions

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With the property that ${\{p_{n+1}^{\prime }(x)\}}_{n=0}^{\mathrm{\infty }}$ is again a system of OP’s. See §18.9(iii).

… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: … ►For proofs of these results and for similar properties of the Chebyshev polynomials of the second, third, and fourth kinds see Mason and Handscomb (2003, §4.6). ►For another version of the discrete orthogonality property of the polynomials $T_n\left(x\right)$ see (3.11.9). … ►In consequence, additional properties are included in Chapter 14. …

7: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

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28.12.2 $${\lambda }_{\nu }\left(-q\right)={\lambda }_{\nu }\left(q\right)={\lambda }_{-\nu }\left(q\right).$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E2

Encodings:

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See also:

Annotations for §28.12(i), §28.12 and Ch.28

… ►They have the following pseudoperiodic and orthogonality properties: … ►

28.12.10 $$\overline{{\mathrm{me}}_{\nu }(z,q)}={\mathrm{me}}_{\overline{\nu }}(-\overline{z},\overline{q}).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\overline{z}$: complex conjugate, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.12(ii)

Permalink:

http://dlmf.nist.gov/28.12.E10

Encodings:

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See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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§28.12(iii) Functions ${\mathrm{ce}}_{\nu }(z,q)$, ${\mathrm{se}}_{\nu }(z,q)$, when $\nu \notin \mathbb{Z}$

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8: 10.41 Asymptotic Expansions for Large Order

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§10.41(iv) Double Asymptotic Properties

… ►Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with $z$ replaced by $\nu z$, up to and including the term in $z^{-(\mathrm{\ell }-1)}$. … ►

§10.41(v) Double Asymptotic Properties (Continued)

… ►We first prove that for the expansions (10.20.6) for the Hankel functions $H_{\nu }^{(1)}\left(\nu z\right)$ and $H_{\nu }^{(2)}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to \pm \mathrm{i}\mathrm{\infty }$, respectively. …We then extend the validity of this property from $z\to \pm \mathrm{i}\mathrm{\infty }$ to $z\to \mathrm{\infty }$ in the sector $-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$ in the case of $H_{\nu }^{(1)}\left(\nu z\right)$, and to $z\to \mathrm{\infty }$ in the sector $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$ in the case of $H_{\nu }^{(2)}\left(\nu z\right)$. …

9: 4.30 Elementary Properties

§4.30 Elementary Properties

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10: 11.8 Analogs to Kelvin Functions

… ►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).