# properties

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## 1—10 of 193 matching pages

##### 1: 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).
##### 2: 22 Jacobian Elliptic Functions
He was also editor of The Kowalevski Property, CRM Proceedings and Lecture Notes 32, published by the American Mathematical Society in 2002. …
##### 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: 28.12 Definitions and Basic Properties
###### §28.12 Definitions and Basic Properties
28.12.2 $\lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).$
They have the following pseudoperiodic and orthogonality properties: …
##### 7: 18.3 Definitions
In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_{n}\left(x\right)$, $n=0,1,\dots,N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\dots,N+1$, 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.
##### 8: 10.41 Asymptotic Expansions for Large Order
###### §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 $z^{-(\ell-1)}$. …
###### §10.41(v) Double Asymptotic Properties (Continued)
We first prove that for the expansions (10.20.6) for the Hankel functions ${H^{(1)}_{\nu}}\left(\nu z\right)$ and ${H^{(2)}_{\nu}}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to\pm i\infty$, respectively. …We then extend the validity of this property from $z\to\pm i\infty$ to $z\to\infty$ in the sector $-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta$ in the case of ${H^{(1)}_{\nu}}\left(\nu z\right)$, and to $z\to\infty$ in the sector $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$ in the case of ${H^{(2)}_{\nu}}\left(\nu z\right)$. …
##### 10: 11.8 Analogs to Kelvin Functions
For properties of Struve functions of argument $xe^{\pm 3\pi i/4}$ see McLachlan and Meyers (1936).