9
ϑ Functions
Abstract
Focus of this chapter are the four Jacobi or classical ϑ functions, Dedekind’s
η function, and the Jacobi symbol. The evaluation of the Jacobi ϑ functions is
based on a Fourier series expansion, optimized by taking their quasi periodic
behavior on a lattice into account and various coordinate transformations. A
similar strategy is used for evaluating Dedekind’s η function. The corresponding
m-files are available for download.
Jacobi or classical ϑ functions play an important rôle in constructing elliptic and
modular functions. ϑ functions are the elliptic analogs of the exponential function.
ϑ functions have been applied to soliton theory and diffusion equations. As an
additional minor topic, we will briefly discuss Dedekind’s η function. Applications
for the η function can be found, e.g., in string theory.
9.1
Function Overview
Jacobi ϑ functions The SPECFUNPHYS class jacobiTheta allows the evaluation of the ϑ functions ϑ 1 , ϑ 2 , ϑ 3 , and ϑ 4 .
Dedekind’s η function The SPECFUNPHYS class dedeEta returns the value of
Dedekind’s η function.
Jacobi symbol The function jacobisymbol returns the Jacobi symbol
a
n
.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_9
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