12.3 The Class ellipWeier
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In case, the Weierstraß functions ℘ (z) and ζ(z) should not converge due to a
high z-value, this z-value will be mapped on the fundamental period parallelogram
and the result of the ζ function evaluated based on its quasiperiodicity.
12.3 The Class ellipWeier
The class classWeier serves for the evaluation of the Weierstraß elliptic functions and some lattice related functions. The syntax is obj = ellipWeier(wweier, omega1, omega3, z). The input variable “wweier” supports the
values:
• “wp”—Weierstraß ℘ (z)-function obj = ellipWeier(’wp’, omega1,
omega3, z);
• “zeta”—Weierstraß ζ(z)-function obj = ellipWeier(’zeta’, omega1,
omega3, z);
• “sigma”—Weierstraß σ (z)-function obj = ellipWeier(’sigma’,
omega1, omega3, z);
• “e1,” “e2,” “e3,” and “eall” for the computation of the zeros of the Weierstraß
cubic normal e i , e.g., obj = ellipWeier(’e1’, omega1, omega3);
• “eta1,” “eta2,” “eta3,” and “etaall,” the quasiperiodic contribution η i to the
ζ(z)-function, e.g., obj=ellipWeier(’etaall’,omega1,omega3) to
compute all three values;
• “g23” to compute the lattice invariants g 2 and g 3 and “disc” for the discriminant
Δ, and g2 and g3;
• “lambda” for the elliptic modular function λ(τ ), obj=ellipWeier(’lambda’, omega1, omega3) or obj=ellipWeier(’lambda’,1,tau);
• “klein” to evaluate Klein’s complete invariant J (τ ), e.g.,
obj = ellipWeier(’klein’, 1, tau); and
• the auxiliary function ℘ (z) − e i via “wpe1,” “wpe2,” “wpe3,” and “wpeall.”
The inputs “omega1” and “omega3” are the half-period lattice generator, with
(ω3/ω1) > 0. Some functions depend only on τ =
ω3
ω1 , and thus one
could also use obj = ellipWeier(wweier, 1, tau) instead of obj =
ellipWeier(wweier, omega1, omega3).
The last input variable z is the complex function variable. All variables have to be
scalars. The output is the object “obj” with property “value” for the evaluation result.
This could be a single number or a vector. “z” is the input at which the function
will be evaluated. In case of ℘ (z) and ζ(z), “z” will have two components if the
evaluation was based on the periodic, respectively, and quasiperiodic properties of
these functions. “para” is the used parameter for evaluation, thus in dependence of
the functions, ω 1 and ω 3 and/or τ . Finally, “info” carries some general information
about the function in quest.
157
In case, the Weierstraß functions ℘ (z) and ζ(z) should not converge due to a
high z-value, this z-value will be mapped on the fundamental period parallelogram
and the result of the ζ function evaluated based on its quasiperiodicity.
12.3 The Class ellipWeier
The class classWeier serves for the evaluation of the Weierstraß elliptic functions and some lattice related functions. The syntax is obj = ellipWeier(wweier, omega1, omega3, z). The input variable “wweier” supports the
values:
• “wp”—Weierstraß ℘ (z)-function obj = ellipWeier(’wp’, omega1,
omega3, z);
• “zeta”—Weierstraß ζ(z)-function obj = ellipWeier(’zeta’, omega1,
omega3, z);
• “sigma”—Weierstraß σ (z)-function obj = ellipWeier(’sigma’,
omega1, omega3, z);
• “e1,” “e2,” “e3,” and “eall” for the computation of the zeros of the Weierstraß
cubic normal e i , e.g., obj = ellipWeier(’e1’, omega1, omega3);
• “eta1,” “eta2,” “eta3,” and “etaall,” the quasiperiodic contribution η i to the
ζ(z)-function, e.g., obj=ellipWeier(’etaall’,omega1,omega3) to
compute all three values;
• “g23” to compute the lattice invariants g 2 and g 3 and “disc” for the discriminant
Δ, and g2 and g3;
• “lambda” for the elliptic modular function λ(τ ), obj=ellipWeier(’lambda’, omega1, omega3) or obj=ellipWeier(’lambda’,1,tau);
• “klein” to evaluate Klein’s complete invariant J (τ ), e.g.,
obj = ellipWeier(’klein’, 1, tau); and
• the auxiliary function ℘ (z) − e i via “wpe1,” “wpe2,” “wpe3,” and “wpeall.”
The inputs “omega1” and “omega3” are the half-period lattice generator, with
(ω3/ω1) > 0. Some functions depend only on τ =
ω3
ω1 , and thus one
could also use obj = ellipWeier(wweier, 1, tau) instead of obj =
ellipWeier(wweier, omega1, omega3).
The last input variable z is the complex function variable. All variables have to be
scalars. The output is the object “obj” with property “value” for the evaluation result.
This could be a single number or a vector. “z” is the input at which the function
will be evaluated. In case of ℘ (z) and ζ(z), “z” will have two components if the
evaluation was based on the periodic, respectively, and quasiperiodic properties of
these functions. “para” is the used parameter for evaluation, thus in dependence of
the functions, ω 1 and ω 3 and/or τ . Finally, “info” carries some general information
about the function in quest.