12
Weierstraß Functions
Abstract
The chapter’s target are the Weierstraß functions ℘ (z, τ ), ζ(z, τ ), and σ (z, τ ).
In addition, several related functions, e.g., the lattice roots, the modular function,
or Klein’s complete invariant, are also discussed. m-Code for all functions is
available for download.
The chapter’s target are the Weierstraß functions ℘ (z, τ ), ζ(z, τ ), and σ (z, τ ).
Applications are, e.g., a point particle in classical dynamics, astrophysical applications, periodic potentials in quantum dynamics, or supra-conductivity to list only
a few.
This chapter completes the discussion of elliptic functions and integrals. The
following equations are based on [1].
12.1 Function Overview
The SPECFUNPHYS class ellipWeier supports the evaluation of
• the Weierstraß elliptic functions ℘ (z), ζ(z), and σ (z),
• the quasiperiodic contribution η i , i = 1, 2, 3 of ζ(z),
• the lattice invariants g2 and g3 and the discriminant Δ,
• the lattice roots e 1 , e 2 , and e 3 ,
• the elliptic modular function λ(τ ) and Klein’s complete invariant J (τ ), and
• some auxiliary functions useful for test purposes ℘ (z) − e i , i = 1, 2, 3.
© 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_12
153
Weierstraß Functions
Abstract
The chapter’s target are the Weierstraß functions ℘ (z, τ ), ζ(z, τ ), and σ (z, τ ).
In addition, several related functions, e.g., the lattice roots, the modular function,
or Klein’s complete invariant, are also discussed. m-Code for all functions is
available for download.
The chapter’s target are the Weierstraß functions ℘ (z, τ ), ζ(z, τ ), and σ (z, τ ).
Applications are, e.g., a point particle in classical dynamics, astrophysical applications, periodic potentials in quantum dynamics, or supra-conductivity to list only
a few.
This chapter completes the discussion of elliptic functions and integrals. The
following equations are based on [1].
12.1 Function Overview
The SPECFUNPHYS class ellipWeier supports the evaluation of
• the Weierstraß elliptic functions ℘ (z), ζ(z), and σ (z),
• the quasiperiodic contribution η i , i = 1, 2, 3 of ζ(z),
• the lattice invariants g2 and g3 and the discriminant Δ,
• the lattice roots e 1 , e 2 , and e 3 ,
• the elliptic modular function λ(τ ) and Klein’s complete invariant J (τ ), and
• some auxiliary functions useful for test purposes ℘ (z) − e i , i = 1, 2, 3.
© 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_12
153