Borel Sum Calculator
FinitepolynomialBorel-Laplace identityΣanx^n:integralexp(−t)Σan(xt)^n/n!dt=Σanx^n; not a method to certifyBorelsummabilityofdivergentinfiniteinputseries. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
For a finite polynomial, the Borel-Laplace identity recovers the polynomial evaluation.
For a finite polynomial, the Borel-Laplace identity recovers the polynomial evaluation.1
This tool evaluates that polynomial without certifying summability of a divergent infinite series.
When to use Borel Sum Calculator
Use this tool to evaluate that polynomial without certifying summability of a divergent infinite series.
How to use Borel Sum Calculator and how it works
- Prepare actualFinitePowerSeriesCoefficients, actualRealEvaluationPoint, with the units, conventions and assumptions below.
- Enter the required fields and run Borel Sum Calculator.
- Read result.exactPolynomialBorelLaplaceValue using the output definitions below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualFinitePowerSeriesCoefficients | Nonempty [a0,a1,...] in ascending-power order, at most 64 finite entries. | array of number | Required; other constraints below |
| actualRealEvaluationPoint | Actual Real Evaluation Point. Use the units and matching context stated in the calculation below. | number | Required; finite; other constraints below |
Require a nonempty finite coefficient array with at most 64 entries and finite x with absolute value at most 10. Treat coefficients as ascending powers and use Horner evaluation from the last coefficient to the first. Return the ordinary finite polynomial value; the output key's word exact identifies the finite-polynomial identity and does not make binary64 arithmetic exact. The tool does not continue or sum a divergent infinite series.
value=a_0+a_1*x+...+a_n*x^n 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.exactPolynomialBorelLaplaceValue | Finite-polynomial numerical value; the key does not certify exact binary64 arithmetic. | number |
{
"input": {
"actualFinitePowerSeriesCoefficients": [
1,
2,
3
],
"actualRealEvaluationPoint": 2
},
"output": {
"result": {
"exactPolynomialBorelLaplaceValue": 17
}
}
}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 Euler Product Approximation for a complementary operation: The zeta Euler product combines factors indexed by primes on a convergent real-exponent domain.
Use Bessel Y0 Approximation for a complementary operation: The order-zero Bessel Y function on the positive real axis includes logarithmic and series terms.
References
-
Borel Sum Calculator — tool.io
Similar or alternative tools
- Euler Product Approximation
The zeta Euler product combines factors indexed by primes on a convergent real-exponent domain.
- Bessel Y0 Approximation
The order-zero Bessel Y function on the positive real axis includes logarithmic and series terms.