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Gamma

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Gamma[z]
    is the Euler gamma function Gamma(z).
Gamma[a, z]
    is the upper incomplete gamma function Gamma(a, z).
Gamma[a, z0, z1]
    is the generalized incomplete gamma Gamma(a, z0) - Gamma(a, z1).
Integer and half-integer arguments reduce to exact values ((z-1)!,
and rational multiples of Sqrt[Pi]); non-positive integers give
ComplexInfinity. Machine and arbitrary-precision (MPFR) real inputs
evaluate numerically, as do machine-precision complex inputs. Listable.

Examples

No verified examples yet for this function.

Implementation notes

  • Exact reductions for Gamma[z]:
  • Positive integers: Gamma[n] = (n-1)! (exact, with GMP BigInt for large n).
  • Non-positive integers are poles: Gamma[0], Gamma[-n] → ComplexInfinity.
  • Half-integers reduce to rational multiples of Sqrt[Pi], e.g.

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Gamma[5]
Out[1]= 24

In[1]:= Gamma[7/2]
Out[1]= 15/8 Sqrt[Pi]

In[1]:= Gamma[-1/2]
Out[1]= -2 Sqrt[Pi]

In[1]:= N[Gamma[1/3], 40]
Out[1]= 2.6789385347077476336556929409746776441289

In[1]:= N[Gamma[3 + 4 I], 20]
Out[1]= 0.00522553847136921419473 - 0.172547079294300187719*I

Notes

Gamma is the Euler gamma function, the analytic continuation of the factorial: Gamma[n] = (n-1)!, so Gamma[5] = 24. Half-integer arguments collapse to exact rational multiples of Sqrt[Pi] — Gamma[7/2] = 15/8 Sqrt[Pi] — and this continues through the poles at non-positive integers into negative half-integers, where Gamma[-1/2] = -2 Sqrt[Pi]. For arguments with no closed form it evaluates to arbitrary precision via MPFR (Gamma[1/3] to 40 digits) and across the complex plane (Gamma[3 + 4 I]). The two- and three-argument forms give the upper incomplete and generalized incomplete gamma functions.