Weierstrass Zeta Approximation

Finite symmetricsquareWeierstrasszetalatticesum/canonical product, actualTWO full periods not half periods, cutoff≤10, poles rejected, 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 zeta function is defined relative to a two-dimensional complex period lattice.

The Weierstrass zeta function is defined relative to a two-dimensional complex period lattice.^[1]

This tool evaluates a finite symmetric-square lattice sum using full periods and returns complex components without a certified infinite-tail estimate.

When to use Weierstrass Zeta Approximation

This tool evaluates a finite symmetric-square lattice sum using full periods and returns complex components without a certified infinite-tail estimate.

How to use Weierstrass Zeta 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 Zeta 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

json

{
  "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 Sigma Approximation for a complementary operation: The Weierstrass sigma function has a canonical lattice-product representation with zeros at lattice points.

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

Weierstrass Zeta Approximation — tool.io
NIST DLMF: Weierstrass definitions and lattices — NIST

Similar or alternative tools

Weierstrass Sigma Approximation
The Weierstrass sigma function has a canonical lattice-product representation with zeros at lattice points.
Weierstrass Elliptic P Approximation
The Weierstrass elliptic p function depends on a complex lattice and has poles at lattice points.