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31: 21.7 Riemann Surfaces

… ►Since a Riemann surface $\mathrm{\Gamma }$ is a two-dimensional manifold that is orientable (owing to its analytic structure), its only topological invariant is its genus $g$ (the number of handles in the surface). …

32: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►The $\mathit{6}j$ symbol is invariant under interchange of any two columns and also under interchange of the upper and lower arguments in each of any two columns, for example, …

33: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►Let $f:𝛀\to ℂ$ (a) be orthogonally invariant, so that $f(𝐓)$ is a symmetric function of $t_1,\mathrm{\dots },t_m$, the eigenvalues of the matrix argument $𝐓\in 𝛀$; (b) be analytic in $t_1,\mathrm{\dots },t_m$ in a neighborhood of $𝐓=𝟎$; (c) satisfy $f(𝟎)=1$. …

34: Bibliography B

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Irregular primes and cyclotomic invariants to 12 million. J. Symbolic Comput. 31 (1-2), pp. 89–96.

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Notes:

Computational algebra and number theory (Milwaukee, WI, 1996)

External Links:

ISSN 0747-7171, Document, MathReview (Xavier-François Roblot), ZentralBlatt

Referenced by:

3rd item.

Encodings:

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35: Bibliography M

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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External Links:

ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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36: Errata

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The statements “If $c$ and $d$ are real” and “If $c$ and $d$ are not both real” have been further clarified (suggested by Alan Barnes on 2021-03-26).

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The Weierstrass lattice roots $e_j,$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_j$, and lattice invariants $g_2$, $g_3$, now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

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37: 1.9 Calculus of a Complex Variable

… ►or its limiting form, and is invariant under bilinear transformations. …

38: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►This dilatation transformation, which does require analyticity of $q(x)$ in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of $⟨{\left(z-T\right)}^{-1}f,f⟩$. …