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.

Description

Upper incomplete gamma measures the tail of the gamma integral beyond x.

Upper incomplete gamma measures the tail of the gamma integral beyond x. Dividing by complete gamma gives the regularized fraction Q, a different quantity from the unnormalized integral.^{[1, 2]}

When to use Upper Incomplete Gamma Approximation

Use this tool to 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.

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

How to use Upper 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 upperIncompleteGammaValue, regularizedUpperGammaFraction. 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]. For x<a+1 compute a convergent lower series and complement; otherwise evaluate the upper continued fraction directly. Stop at relative criterion 10⁻¹⁴ within 10,000 iterations, reject nonconvergence and return both upper quantities. At x=0, Q=1.

Γ(a,x)=Γ(a)Q(a,x)

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

Output field	Meaning	Value type
upperIncompleteGammaValue	Upper incomplete gamma value	number
regularizedUpperGammaFraction	Regularized upper gamma fraction	number

json

{
  "input": {
    "shape": 2,
    "x": 1
  },
  "output": {
    "upperIncompleteGammaValue": 0.7357588823428848,
    "regularizedUpperGammaFraction": 0.7357588823428848
  }
}

Limitations and assumptions

The implementation is floating-point and has no certified total-error bound. Complements near an endpoint can lose tiny-tail resolution.

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 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. 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

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

Similar or alternative tools

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.
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.