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21 Multidimensional Theta FunctionsNotation

§21.1 Special Notation

(For other notation see Notation for the Special Functions.)

g,h positive integers.
g ××× (g times).
g ××× (g times).
g×h set of all g×h matrices with integer elements.
𝛀 g×g complex, symmetric matrix with 𝛀 strictly positive definite, i.e., a Riemann matrix.
𝜶,𝜷 g-dimensional vectors, with all elements in [0,1), unless stated otherwise.
aj jth element of vector 𝐚.
Ajk (j,k)th element of matrix 𝐀.
𝐚𝐛 scalar product of the vectors 𝐚 and 𝐛.
𝐚𝛀𝐛 [𝛀𝐚]𝐛=[𝛀𝐛]𝐚.
diag𝐀 Transpose of [A11,A22,,Agg].
𝟎g g×g zero matrix.
𝐈g g×g identity matrix.
𝐉2g [𝟎g𝐈g𝐈g𝟎g].
Sg set of g-dimensional vectors with elements in S.
|S| number of elements of the set S.
S1S2 set of all elements of the form “element of S1×element of S2”.
S1/S2 set of all elements of S1, modulo elements of S2. Thus two elements of S1/S2 are equivalent if they are both in S1 and their difference is in S2. (For an example see §20.12(ii).)
ab intersection index of a and b, two cycles lying on a closed surface. ab=0 if a and b do not intersect. Otherwise ab gets an additive contribution from every intersection point. This contribution is 1 if the basis of the tangent vectors of the a and b cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is 1.
aω line integral of the differential ω over the cycle a.

Lowercase boldface letters or numbers are g-dimensional real or complex vectors, either row or column depending on the context. Uppercase boldface letters are g×g real or complex matrices.

The main functions treated in this chapter are the Riemann theta functions θ(𝐳|𝛀), and the Riemann theta functions with characteristics θ[𝜶𝜷](𝐳|𝛀).

The function Θ(ϕ|𝐁)=θ(ϕ/(2πi)|𝐁/(2πi)) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).