Lower Incomplete Gamma Approximation

Evaluate lower integral γ(a,x) and P separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.

Description

Lower incomplete gamma integrates from zero to x.

Lower incomplete gamma integrates from zero to x. Its regularized fraction P removes the complete-gamma normalization and must not be confused with the integral itself.^{[1, 2]}

When to use Lower Incomplete Gamma Approximation

Use this tool to evaluate lower integral γ(a,x) and P separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Lower Incomplete Gamma 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 shape, x using the field table and calculation rules, then run the tool.
Read lowerIncompleteGammaValue, regularizedLowerGammaFraction. 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
shape	Shape	number	Required
x	X	number	Required

Use 0<a≤170 and x in [0,10,000]. Use the lower power series when x<a+1 and the complemented upper continued fraction otherwise; both iterations are bounded to 10,000 steps with relative criterion 10⁻¹⁴. Return normalized and unnormalized values; x=0 returns both zero.

γ(a,x)=Γ(a)P(a,x)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
lowerIncompleteGammaValue	Lower incomplete gamma value	number
regularizedLowerGammaFraction	Regularized lower gamma fraction	number

json

{
  "input": {
    "shape": 2,
    "x": 1
  },
  "output": {
    "lowerIncompleteGammaValue": 0.2642411176571152,
    "regularizedLowerGammaFraction": 0.2642411176571152
  }
}

Limitations and assumptions

No arbitrary-precision or certified error result is supplied. Floating endpoint complements can lose small residuals.

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 Upper Incomplete Gamma Approximation: Evaluate upper tail Γ(a,x) and Q separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused. 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.

Further background: Incomplete gamma functions: definitions.

References

Lower Incomplete Gamma Approximation — tool.io
Incomplete gamma functions: definitions — NIST Digital Library of Mathematical Functions

Similar or alternative tools

Upper Incomplete Gamma Approximation
Evaluate upper tail Γ(a,x) and Q separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.
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.