Rogers-Ramanujan Continued Fraction Approximation
Rogers–Ramanujancontinuedfractionq^(1/5)/(1+q/(1+q²/(…))), finite explicit depth, no certified infinite convergence error. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
The Rogers-Ramanujan continued fraction has an infinite nested structure and a specified leading q factor.
The Rogers-Ramanujan continued fraction has an infinite nested structure and a specified leading q factor.1
This tool evaluates a finite backward recurrence on a real q branch without certifying the omitted depth.
When to use Rogers-Ramanujan Continued Fraction Approximation
This tool evaluates a finite backward recurrence on a real q branch without certifying the omitted depth.
How to use Rogers-Ramanujan Continued Fraction Approximation and how it works
- Prepare q, depth, using the input definitions and matching units below.
- Enter the required fields and run Rogers-Ramanujan Continued Fraction Approximation.
- Read result.truncatedContinuedFraction 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 |
|---|---|---|---|
| actualQ | Real series or continued-fraction variable q in the tool-specific branch stated below. | number | Required; fraction from 0 to 1; finite; other constraints below |
| actualDepth | Integer finite recurrence or product depth under the stated cutoff. | number | Required; integer 1–256; finite; other constraints below |
Require real q strictly between 0 and 1 and depth integer 1–256. Set denominator to one, then update denominator=1+q^k/denominator for k from depth down to one. Return q^(1/5)/denominator as the truncated continued fraction. No infinite convergence bound or complex branch is supplied.
d=1; for k=N..1:d=1+q^k/d; result=q^(1/5)/d 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.truncatedContinuedFraction | Finite Rogers-Ramanujan continued fraction under the stated branch and truncation; no additional numerical certificate. | number |
{
"input": {
"actualQ": 0.25,
"actualDepth": 1
},
"output": {
"result": {
"truncatedContinuedFraction": 0.6062866266041592
}
}
}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.
Alternative or Complementary methods
Use Mock Theta Function Approximation for a complementary operation: Ramanujan's third-order mock theta f is one named function within a broader family.
Use Modular Discriminant Delta Calculator for a complementary operation: The modular discriminant has a q-product whose finite truncation omits later factors.
References
Similar or alternative tools
- Mock Theta Function Approximation
Ramanujan's third-order mock theta f is one named function within a broader family.
- Modular Discriminant Delta Calculator
The modular discriminant has a q-product whose finite truncation omits later factors.