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SPECIAL FUNCTIONS / LOG-GAMMA DERIVATIVE

Real Digamma Function Calculator

Evaluate the positive-real psi function with recurrence and special-value context.

  • 01 Calculated in this tab
  • 02 Values stay in this browser tab
  • 03 Use boundary

Conversion input

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Function inputs

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METHOD / WORKED EXAMPLE

Check the log-Gamma derivative through recurrence

Digamma, also called psi, is the derivative of ln Gamma. The recurrence check distinguishes it from Gamma values and makes the approximation testable at the entered argument.

WORKED DEFAULT

Check the calculation with the default inputs

At x = 1, psi(1) equals negative Euler's constant, approximately -0.5772156649. The recurrence predicts psi(2) = psi(1) + 1, approximately 0.4227843351.

  1. Use the special pointpsi(1) = -gamma
  2. Insert Euler's constantpsi(1) = -0.5772156649...
  3. Check recurrencepsi(2) = psi(1) + 1

READ THE RESULT

Interpret the output in context

The value describes the local rate of change of log Gamma. It is not a finite difference, probability, or Gamma value, and it approaches logarithmic behavior only for large arguments.

ASSUMPTIONS AND LIMITS

Know where the model stops

  • x is positive and real.
  • The ordinary digamma function is intended rather than a higher polygamma derivative.
  • The required precision fits double-precision asymptotic evaluation.

This page evaluates the first derivative of log Gamma, not Gamma itself, and excludes complex values and nonpositive poles.

NIST DLMF: Gamma and psi definitions

COMMON QUESTIONS

Real Digamma Function Calculator FAQs

Is digamma the same as the derivative of Gamma?

Not exactly. Digamma is the logarithmic derivative Gamma'(x)/Gamma(x), which is also the derivative of ln Gamma(x). Gamma's ordinary derivative equals Gamma(x) multiplied by psi(x). Keeping that normalization matters because digamma grows roughly like ln(x), while Gamma and its direct derivative can grow extraordinarily quickly numerically.

Why does psi(1) involve Euler's constant?

A standard special value is psi(1) = -gamma, where gamma is the Euler-Mascheroni constant. This gives a strong numerical checkpoint for an implementation and anchors the recurrence psi(x+1) = psi(x) + 1/x. Euler's constant here is unrelated to the Gamma function's name despite the shared symbol convention.

Can this evaluate digamma at zero or negative integers?

No. Digamma has poles at zero and the negative integers, so no finite value exists there. Analytic continuation can evaluate many negative non-integers, but reflection formulas and near-pole behavior require separate numerical handling. This page keeps a positive-real domain so its recurrence, asymptotic method, and displayed residual remain straightforward.

Use boundary

Calculation path

Shift the positive argument upward with the digamma recurrence, evaluate a Bernoulli asymptotic series in a stable range, and shift the accumulated terms back.

Calculation path

psi(x) = derivative of ln Gamma(x); psi(x+1) = psi(x) + 1/x.