Mock Theta Function Approximation

ExplicitRamanujanthirdordermocktheta f(q)=Σq^(n²)/Π(1+q^k)², termsn=0…N−1, not unnamed family or analytic continuation. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.

Description

Ramanujan's third-order mock theta f is one named function within a broader family.

Ramanujan's third-order mock theta f is one named function within a broader family.^[1]

This tool sums its specified finite real-q series without evaluating an unspecified mock theta function or analytic continuation.

When to use Mock Theta Function Approximation

This tool sums its specified finite real-q series without evaluating an unspecified mock theta function or analytic continuation.

How to use Mock Theta Function Approximation and how it works

Prepare q, term budget, using the input definitions and matching units below.
Enter the required fields and run Mock Theta Function Approximation.
Read result.truncatedThirdOrderMockThetaF 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; finite; other constraints below
actualTermBudget	Number of included terms under the stated series indexing convention.	number	Required; integer 1–256; finite; other constraints below

Require finite real q with absolute value below one and term budget integer 1–256. Include the n=0 term one, accumulate product(1+q^k) through each n, and add q^(n^2) divided by that product squared through n=budget-1. Return the truncated third-order mock theta f; q=0 returns one. No omitted-tail bound or complex continuation occurs.

f_N(q)=sum(q^(n^2)/product((1+q^k)^2,k=1..n),n=0..N-1)

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.truncatedThirdOrderMockThetaF	Finite third-order mock theta f series under the stated branch and truncation; no additional numerical certificate.	number

json

{
  "input": {
    "actualQ": 0.5,
    "actualTermBudget": 2
  },
  "output": {
    "result": {
      "truncatedThirdOrderMockThetaF": 1.2222222222222223
    }
  }
}

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 Rogers-Ramanujan Continued Fraction Approximation for a complementary operation: The Rogers-Ramanujan continued fraction has an infinite nested structure and a specified leading q factor.

Use Modular Discriminant Delta Calculator for a complementary operation: The modular discriminant has a q-product whose finite truncation omits later factors.

References

Mock Theta Function Approximation — tool.io

Similar or alternative tools

Rogers-Ramanujan Continued Fraction Approximation
The Rogers-Ramanujan continued fraction has an infinite nested structure and a specified leading q factor.
Modular Discriminant Delta Calculator
The modular discriminant has a q-product whose finite truncation omits later factors.