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: 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. …

8: 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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FullText, MathReview (T. Oda), ZentralBlatt

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§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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§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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Single-precision Fortran, accuracy 12-14S

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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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MathReview (R. A. Askey), ZentralBlatt

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§18.37(ii).

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9: 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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10: 22.19 Physical Applications

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§22.19(i) Classical Dynamics: The Pendulum

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§22.19(ii) Classical Dynamics: The Quartic Oscillator

… ►The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). … ►

§22.19(v) Other Applications

►Numerous other physical or engineering applications involving Jacobian elliptic functions, and their inverses, to problems of classical dynamics, electrostatics, and hydrodynamics appear in Bowman (1953, Chapters VII and VIII) and Lawden (1989, Chapter 5). …