general properties

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31: 22 Jacobian Elliptic Functions

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32: 21.3 Symmetry and Quasi-Periodicity

… ►Because of this property, the elements of $𝜶$ and $𝜷$ are usually restricted to $[0,1)$, without loss of generality. …

33: 15.2 Definitions and Analytical Properties

… ►The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. …

34: 21.5 Modular Transformations

… ►For general $𝚪$, it is difficult to decide which root needs to be used. …Equation (21.5.4) is the modular transformation property for Riemann theta functions. ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi (𝚪)$ is determinate: …

35: 8.2 Definitions and Basic Properties

§8.2 Definitions and Basic Properties

… ►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by … ►In this subsection the functions $\gamma$ and $\mathrm{\Gamma }$ have their general values. …

36: 18.27 $q$-Hahn Class

… ►The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters. ►All these systems of OP’s have orthogonality properties of the form …In case of unbounded sequences (18.27.2) can be rewritten as a $q$-integral, see §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. … ►They are defined by their $q$-hypergeometric representations, followed by their orthogonality properties. …

37: 19.15 Advantages of Symmetry

… ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. … ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

38: 28.29 Definitions and Basic Properties

§28.29 Definitions and Basic Properties

… ►A generalization of Mathieu’s equation (28.2.1) is Hill’s equation … ►Let $\nu$ be a real or complex constant satisfying (without loss of generality) …Then (28.29.1) has a nontrivial solution $w(z)$ with the pseudoperiodic property …

39: 23.2 Definitions and Periodic Properties

§23.2 Definitions and Periodic Properties

… ► … ►In general, if … ►

§23.2(ii) Weierstrass Elliptic Functions

… ►For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).

40: 28.12 Definitions and Basic Properties

§28.12 Definitions and Basic Properties

… ►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict $\widehat{\nu }\ne 0,1$; equivalently $\nu \ne n$. … ►Without loss of generality, from now on we replace $\nu +2n$ by $\nu$. … ►They have the following pseudoperiodic and orthogonality properties: … ►

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

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