Dedekind Eta Approximation

Approximate real positive-nome Dedekind eta q^(1/24)∏_(n=1..N)(1−q^n), τ purely imaginary and q=exp(2πiτ); explicit finite product, not generic complex eta or a duplicate alias.

Description

Dedekind eta contains both a fractional nome prefactor and an infinite product.

Dedekind eta contains both a fractional nome prefactor and an infinite product. A positive real nome corresponds to a purely imaginary modular argument under q=exp(2πiτ).^[1]

Approximate real positive-nome Dedekind eta q^(1/24)∏_(n=1..N)(1−q^n), τ purely imaginary and q=exp(2πiτ); explicit finite product, not generic complex eta or a duplicate alias.

When to use Dedekind Eta Approximation

Use this tool to approximate real positive-nome Dedekind eta q^(1/24)∏_(n=1..N)(1−q^n), τ purely imaginary and q=exp(2πiτ); explicit finite product, not generic complex eta or a duplicate alias.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Dedekind Eta Approximation and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter realNome, productFactors using the field table and calculation rules, then run the tool.
Read finiteDedekindEtaRealNome. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
realNome	Real nome	number	Required
productFactors	Product factors	number	Required

Use real q strictly between 0 and 1 and integer factors 1–100,000. Add log(q)/24 and stable log1p factor terms, then exponentiate to return the finite real-nome product.

finite eta = q^(1/24)∏[k=1..N](1−q^k)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
finiteDedekindEtaRealNome	Finite dedekind eta real nome	number

json

{
  "input": {
    "realNome": 0.1,
    "productFactors": 3
  },
  "output": {
    "finiteDedekindEtaRealNome": 0.8086796707457473
  }
}

Limitations and assumptions

Complex modular arguments and omitted-tail bounds are excluded. Underflow may round a very small result to zero.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Partition Generating Function Approximation: Evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform. For a complementary task, use Jacobi Theta Approximation: Evaluate specifically real Jacobi θ3(z,q)=1+2Σ q^(n²)cos(2nz) with a geometric upper tail bound; theta index/real-nome convention explicit, not all theta functions or complex continuation.

References

Dedekind Eta Approximation — tool.io

Similar or alternative tools

Partition Generating Function Approximation
Evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform.
Jacobi Theta Approximation
Evaluate specifically real Jacobi θ3(z,q)=1+2Σ q^(n²)cos(2nz) with a geometric upper tail bound; theta index/real-nome convention explicit, not all theta functions or complex continuation.