Confluent Hypergeometric 1F1 Approximation

Compute finite confluent 1F1(a;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.

Description

Confluent hypergeometric 1F1 has one numerator and one denominator parameter.

Confluent hypergeometric 1F1 has one numerator and one denominator parameter. A supplied term count produces a finite series sum rather than automatic convergence certification.^{[1, 2]}

When to use Confluent Hypergeometric 1F1 Approximation

Use this tool to compute finite confluent 1F1(a;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.

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

How to use Confluent Hypergeometric 1F1 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, c, x, seriesTerms using the field table and calculation rules, then run the tool.
Read finiteHypergeometricSum, lastIncludedTermMagnitude, polynomialTerminationObserved. 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
c	C	number	Required
x	X	number	Required
seriesTerms	Series terms	number	Required

Use |a|,|c|≤100, |x|≤20 and total terms 1–1000 including t0. Reject nonpositive integer c. Update terms and flag an encountered zero numerator factor; return sum and last-term magnitude.

t0=1; t_k=t_(k−1)(a+k−1)x/[(c+k−1)k]

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

Output field	Meaning	Value type
finiteHypergeometricSum	Finite hypergeometric sum	number
lastIncludedTermMagnitude	Last included term magnitude	number
polynomialTerminationObserved	Polynomial termination observed	boolean

json

{
  "input": {
    "a": 1,
    "c": 2,
    "x": 0.5,
    "seriesTerms": 4
  },
  "output": {
    "finiteHypergeometricSum": 1.296875,
    "lastIncludedTermMagnitude": 0.005208333333333333,
    "polynomialTerminationObserved": false
  }
}

Limitations and assumptions

The last term is not a total tail bound. Polynomial termination can remain unobserved if the requested count stops too early.

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. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Hypergeometric 2F1 Approximation: Compute finite Gauss 2F1(a,b;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate. For a complementary task, use Struve Function Approximation: Evaluate Struve H_ν real power series for integer order 0..5 and x∈[0,20]; fixed family/order convention, bounded convergence, no ambiguous unnamed special function.

Further background: Confluent hypergeometric functions: definitions.

References

Confluent Hypergeometric 1F1 Approximation — tool.io
Confluent hypergeometric functions: definitions — NIST Digital Library of Mathematical Functions

Similar or alternative tools

Hypergeometric 2F1 Approximation
Compute finite Gauss 2F1(a,b;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.
Struve Function Approximation
Evaluate Struve H_ν real power series for integer order 0..5 and x∈[0,20]; fixed family/order convention, bounded convergence, no ambiguous unnamed special function.