Weierstrass Sigma Approximation
Finite symmetricsquareWeierstrasssigmalatticesum/canonical product, actualTWO full periods not half periods, cutoff≤10, zeros retained; lattice-independent real periods required, no certified full infinite elliptic function. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
The Weierstrass sigma function has a canonical lattice-product representation with zeros at lattice points.
The Weierstrass sigma function has a canonical lattice-product representation with zeros at lattice points.1
This tool evaluates a finite symmetric-square product using two independent full periods without certifying the infinite product.
When to use Weierstrass Sigma Approximation
This tool evaluates a finite symmetric-square product using two independent full periods without certifying the infinite product.
How to use Weierstrass Sigma Approximation and how it works
- Prepare complex z, complex full periods, symmetric lattice cutoff, using the input definitions and matching units below.
- Enter the required fields and run Weierstrass Sigma Approximation.
- Read result.real, result.imaginary using the units, signs and ordering explained below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualComplexZ | Complex argument as [real part, imaginary part]. | array of number | Required; other constraints below |
| actualComplexFullPeriods | Exactly two full complex lattice generators, each [real part, imaginary part]; real-linearly independent. | array of array of number | Required; other constraints below |
| actualSymmetricLatticeCutoff | Square-lattice cutoff N, integer 0–10; indices each range from -N to N. | number | Required; finite; other constraints below |
Supply complex z as [real,imaginary] and exactly two complex full periods in a 2-by-2 matrix. Require a nonzero real determinant of the two periods and lattice cutoff integer 0–10. Enumerate omega=i*period1+j*period2 for -N<=i,j<=N excluding the origin. Apply the finite p or zeta sum or sigma canonical product named by the tool, returning real and imaginary components. The p/zeta branches reject z=0 and sampled poles; sigma retains zeros. Unsampled lattice poles and the omitted infinite tail are not certified. Full periods are supplied, not half-periods.
p_N=1/z^2+sum(1/(z-omega)^2-1/omega^2); zeta_N=1/z+sum(1/(z-omega)+1/omega+z/omega^2); sigma_N=z*product((1-z/omega)*exp(z/omega+z^2/(2*omega^2))) Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.real | Real component of the named finite complex calculation. | number |
| result.imaginary | Imaginary component of the named finite complex calculation. | number |
{
"input": {
"actualComplexZ": [
1,
0
],
"actualComplexFullPeriods": [
[
4,
0
],
[
0,
4
]
],
"actualSymmetricLatticeCutoff": 0
},
"output": {
"result": {
"real": 1,
"imaginary": 0
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
DLMF defines the corresponding infinite lattice sum or product. This implementation uses a bounded symmetric-square cutoff and accepts full period generators rather than half-periods.2
Alternative or Complementary methods
Use Weierstrass Zeta Approximation for a complementary operation: The Weierstrass zeta function is defined relative to a two-dimensional complex period lattice.
Use Weierstrass Elliptic P Approximation for a complementary operation: The Weierstrass elliptic p function depends on a complex lattice and has poles at lattice points.
References
Similar or alternative tools
- Weierstrass Zeta Approximation
The Weierstrass zeta function is defined relative to a two-dimensional complex period lattice.
- Weierstrass Elliptic P Approximation
The Weierstrass elliptic p function depends on a complex lattice and has poles at lattice points.