Airy Bi Approximation
Airy Bi 160-coefficient series|x|≤4, fixed bounded approximation no large argument claim.
Description
Ai and Bi are distinct solutions of Airy's differential equation with different initial values.
Ai and Bi are distinct solutions of Airy's differential equation with different initial values.1
This tool evaluates a bounded real Bi series without extending the approximation to large or complex arguments.
When to use Airy Bi Approximation
Use this tool to evaluate a bounded real Bi series without extending the approximation to large or complex arguments.
How to use Airy Bi Approximation and how it works
- Prepare actualX, with the units, conventions and assumptions below.
- Enter the required fields and run Airy Bi Approximation.
- Read result.value 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 |
|---|---|---|---|
| actualX | Actual X. Use the units and matching context stated in the calculation below. | number | Required; finite; other constraints below |
Require a finite real x with absolute value at most 4. Initialize 160 power-series coefficients with the stored Airy Ai or Bi value and derivative at zero and c_2=0, then set c_(n+3)=c_n/((n+3)*(n+2)). Evaluate the resulting polynomial with Horner's method and return the selected function value. Only the bounded real branch is supported; no arbitrary-precision or rigorous truncation-error guarantee is returned.
y''=x*y; c_(n+3)=c_n/((n+3)*(n+2)); value=sum c_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.value | Numerical value of the named bounded real special-function approximation. | number |
{
"input": {
"actualX": 0
},
"output": {
"result": {
"value": 0.6149266274460007
}
}
}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 Airy equation and the Ai and Bi initial values. This implementation evaluates a bounded finite power series instead of an arbitrary-precision function.2
Alternative or Complementary methods
Use Airy Zero Derivative Approximation for a complementary operation: An Airy derivative-zero reference needs a chosen Ai-prime or Bi-prime family and certified index.
Use Airy Zero Approximation for a complementary operation: An Airy zero reference needs a chosen Ai or Bi family and a certified zero index.
References
Similar or alternative tools
- Airy Zero Derivative Approximation
An Airy derivative-zero reference needs a chosen Ai-prime or Bi-prime family and certified index.
- Airy Zero Approximation
An Airy zero reference needs a chosen Ai or Bi family and a certified zero index.