Beta Function Approximation
Evaluate positive real beta function and stable logarithm from Lanczos log gamma; no probability distribution sampling or continuation across beta poles.
Description
The beta function relates two positive real shape parameters through gamma normalization.
The beta function relates two positive real shape parameters through gamma normalization. Its natural logarithm remains useful when the direct function is very small.1
Evaluate positive real beta function and stable logarithm from Lanczos log gamma; no probability distribution sampling or continuation across beta poles.
When to use Beta Function Approximation
Use this tool to evaluate positive real beta function and stable logarithm from Lanczos log gamma; no probability distribution sampling or continuation across beta poles.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Beta Function 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 a, b using the field table and calculation rules, then run the tool.
- Read betaValue, logBetaValue. 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 |
|---|---|---|---|
| a | A | number | Required |
| b | B | number | Required |
Enter positive real a and b at most 1000. Use Lanczos log-gamma values to compute log beta, then exponentiate for beta; return both.
log B(a,b)=log Γ(a)+log Γ(b)−log Γ(a+b) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| betaValue | Beta value | number |
| logBetaValue | Log beta value | number |
{
"input": {
"a": 2,
"b": 3
},
"output": {
"betaValue": 0.08333333333333355,
"logBetaValue": -2.4849066497879977
}
}Limitations and assumptions
The tool does not sample a beta distribution or continue across nonpositive-argument poles. Direct beta can underflow 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 Incomplete Beta Approximation: Evaluate lower incomplete B_x(a,b) and regularized I_x using symmetry-aware continued fractions, separating normalization; shapes (0,1000], x∈[0,1], bounded iteration with failure refusal. For a complementary task, use Gamma Function Approximation: Evaluate positive real Γ(x) and log Γ(x) by Lanczos/reflection approximation over bounded finite domain; negative poles and complex continuation intentionally excluded.
References
-
Beta Function Approximation — tool.io
Similar or alternative tools
- Incomplete Beta Approximation
Evaluate lower incomplete B_x(a,b) and regularized I_x using symmetry-aware continued fractions, separating normalization; shapes (0,1000], x∈[0,1], bounded iteration with failure refusal.
- Gamma Function Approximation
Evaluate positive real Γ(x) and log Γ(x) by Lanczos/reflection approximation over bounded finite domain; negative poles and complex continuation intentionally excluded.