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SPECIAL FUNCTIONS / CONTINUOUS FACTORIAL

Real Gamma Function Calculator

Evaluate Gamma(x) for a positive real argument with a visible recurrence check.

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

Conversion input

Known value

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Function inputs

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METHOD / WORKED EXAMPLE

Extend factorials without hiding the recurrence

Gamma extends the factorial relationship to positive non-integer arguments. Showing Gamma(x + 1) beside x Gamma(x) provides a useful identity check rather than presenting an isolated approximation.

WORKED DEFAULT

Check the calculation with the default inputs

For x = 5, Gamma(5) = 24. The recurrence gives Gamma(6) = 120, while 5 x Gamma(5) also equals 120; for integers, Gamma(n) = (n - 1)!.

  1. Enter the argumentx = 5
  2. Evaluate GammaGamma(5) = 24
  3. Check recurrenceGamma(6) = 5 x 24 = 120

READ THE RESULT

Interpret the output in context

Integer agreement is a verification landmark, but Gamma also has meaningful non-integer values. The output remains an approximation constrained by the stated positive-real domain and floating-point range.

ASSUMPTIONS AND LIMITS

Know where the model stops

  • x is positive and finite.
  • The required result is real rather than complex.
  • The magnitude fits standard double-precision arithmetic.

This bounded real calculator excludes nonpositive arguments, complex values, poles, and arbitrary-precision evaluation.

NIST DLMF: Gamma and psi definitions

COMMON QUESTIONS

Real Gamma Function Calculator FAQs

Why is Gamma(5) equal to 24 instead of 120?

For positive integers, Gamma(n) equals (n - 1)!, not n!. Therefore Gamma(5) equals 4!, which is 24, while Gamma(6) equals 5!, which is 120. The recurrence Gamma(x + 1) = x Gamma(x) makes this shift explicit and also continues to hold for positive non-integer arguments.

Why does this page stop at x = 170?

Gamma grows extremely quickly. Around larger positive arguments, the direct value exceeds the largest finite JavaScript number even though log Gamma remains representable. The limit prevents an infinity result from looking meaningful. Use the separate log-Gamma calculator when you need stable magnitude comparisons, likelihood calculations, or ratios involving larger arguments.

Can I calculate Gamma for a negative number here?

No. Gamma can be extended to many negative non-integers through analytic continuation, but it has poles at zero and the negative integers and requires careful sign and reflection handling. This page deliberately uses the positive real domain so its integral interpretation, bounds, recurrence check, and numerical behavior remain clear and predictable.

Use boundary

Calculation path

Evaluate the positive-real Gamma function with a Lanczos approximation, then independently expose its logarithm and one-step recurrence identity.

Calculation path

Gamma(x) = integral from 0 to infinity of exp(-t) t^(x-1) dt.