Digamma Approximation

Evaluate real positive digamma ψ(x) using upward recurrence and Bernoulli-corrected large-argument expansion, not finite-difference differentiation or negative-pole continuation.

Description

Digamma is the logarithmic derivative of gamma.

Digamma is the logarithmic derivative of gamma. A recurrence can move small positive arguments into a range suitable for an asymptotic expansion.^{[1, 2]}

Evaluate real positive digamma ψ(x) using upward recurrence and Bernoulli-corrected large-argument expansion, not finite-difference differentiation or negative-pole continuation.

When to use Digamma Approximation

Use this tool to evaluate real positive digamma ψ(x) using upward recurrence and Bernoulli-corrected large-argument expansion, not finite-difference differentiation or negative-pole continuation.

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

How to use Digamma 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 x using the field table and calculation rules, then run the tool.
Read digammaValue. 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
x	X	number	Required

Use x in [0.0001,one million]. Shift the argument upward until at least 10 while accumulating recurrence terms, then apply the Bernoulli-corrected inverse-power expansion.

ψ(x)=Γ′(x)/Γ(x); ψ(x)=ψ(x+1)−1/x

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

Output field	Meaning	Value type
digammaValue	Digamma value	number

json

{
  "input": {
    "x": 1
  },
  "output": {
    "digammaValue": -0.5772156649015527
  }
}

Limitations and assumptions

Negative poles and complex continuation are excluded. No arbitrary-precision or certified error value is returned.

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 Polygamma Approximation: Approximate ψ^(m)(x) for positive x and integer m=1..6 via finite absolutely convergent reciprocal-power series, with integral-test absolute tail upper bound; finite truncation not hidden exact value. 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: Gamma function: definitions.

References

Digamma Approximation — tool.io
Gamma function: definitions — NIST Digital Library of Mathematical Functions

Similar or alternative tools

Polygamma Approximation
Approximate ψ^(m)(x) for positive x and integer m=1..6 via finite absolutely convergent reciprocal-power series, with integral-test absolute tail upper bound; finite truncation not hidden exact value.
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.