McKean and Moll theta functions

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1: 20.1 Special Notation

… ►McKean and Moll’s notation: ${\vartheta }_j(z|\tau )={\theta }_j\left(\pi z\right|\tau )$, $j=1,2,3,4$. …

2: 20.12 Mathematical Applications

§20.12 Mathematical Applications

… ►For applications of ${\theta }_3(0,q)$ to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143). ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). … ►For the terminology and notation see McKean and Moll (1999, pp. 48–53). … ►

3: 20.7 Identities

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§20.7(i) Sums of Squares

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§20.7(ii) Addition Formulas

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§20.7(v) Watson’s Identities

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§20.7(vi) Landen Transformations

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§20.7(vii) Derivatives of Ratios of Theta Functions

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4: 20 Theta Functions

Chapter 20 Theta Functions

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5: 20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

… ►and the notation of §19.2(ii), the complete Legendre integrals of the first kind may be expressed as theta functions: … ►

§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. … ►

§20.9(iii) Riemann Zeta Function

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6: 20.11 Generalizations and Analogs

… ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). …

7: 23.15 Definitions

§23.15 Definitions

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§23.15(i) General Modular Functions

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Elliptic Modular Function

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Dedekind’s Eta Function (or Dedekind Modular Function)

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8: 23.22 Methods of Computation

§23.22 Methods of Computation

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§23.22(i) Function Values

… ►For $\mathrm{\wp }\left(z\right)$ we apply (23.6.2) and (23.6.5), generating all needed values of the theta functions by the methods described in §20.14. … ► … ►The determination of suitable generators $2{\omega }_1$ and $2{\omega }_3$ is the classical inversion problem (Whittaker and Watson (1927, §21.73), McKean and Moll (1999, §2.12); see also §20.9(i) and McKean and Moll (1999, §2.16)). …