classical theta functions

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1: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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2: 21.2 Definitions

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§21.2(iii) Relation to Classical Theta Functions

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3: 21.3 Symmetry and Quasi-Periodicity

… ►See also §20.2(iii) for the case $g=1$ and classical theta functions.

4: 20.13 Physical Applications

… ►Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). …

5: 20.1 Special Notation

… ►Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21. …

6: 21.6 Products

… ►For addition formulas for classical theta functions see §20.7(ii).

7: Victor H. Moll

… ►His interest in Special Functions came later. …Currently he is the scientific editor of the classical table by Gradshteyn and Ryzhik Gradshteyn and Ryzhik (2015). … ►

20 Theta Functions

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23 Weierstrass Elliptic and Modular Functions

8: 18.16 Zeros

… ►Let ${\theta }_{n,m}={\theta }_{n,m}^{(\alpha ,\beta )}$, $m=1,2,\mathrm{\dots },n$, denote the zeros of $P_n^{(\alpha ,\beta )}\left(\cos \theta \right)$ as function of $\theta$ with … ►

Inequalities

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Asymptotic Behavior

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§18.16(vii) Discriminants

►The discriminant (18.2.20) can be given explicitly for classical OP’s. …

9: Bibliography K

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Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.

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§8.24(ii).

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Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

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External Links:

FullText, MathReview (T. Oda), ZentralBlatt

Referenced by:

§21.6(ii).

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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.

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FullText, MathReview, ZentralBlatt

Referenced by:

§8.13(iii), §8.22(ii).

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Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.

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Notes:

Single-precision Fortran, accuracy 12-14S

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1st item.

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Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.

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External Links:

MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.37(ii).

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10: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. … ►

20 Theta Functions

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22 Jacobian Elliptic Functions

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23 Weierstrass Elliptic and Modular Functions

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