Working definition
The gamma function extends the factorial off the integers. DLMF 5.5.1 states the recurrence that fixes its character: Gamma(z+1) equals z times Gamma(z), which reproduces the factorial on the positive integers and defines the function elsewhere by continuation. Two further relations do the work in practice. The reflection formula at 5.5.3, Gamma(z) times Gamma(1-z) equals pi over sin(pi z), converts an argument in one half-plane into one in the other and holds for z not a non-positive integer. The duplication formula at 5.5.5 relates Gamma(2z) to Gamma(z) and Gamma(z+1/2). Large arguments are handled by the Stirling expansion at 5.11.1, valid as z tends to infinity in the sector where the phase of z is at most pi minus delta.
Notation
Gamma(z)psi(z) for the digamma functionB_2k for the Bernoulli numbers appearing in the Stirling seriesAssumptions
- The reflection formula at 5.5.3 requires z to be neither zero nor a negative integer.
- The duplication formula at 5.5.5 requires 2z to be neither zero nor a negative integer, which is a stricter condition than the one on 5.5.3.
- The Stirling expansion at 5.11.1 is stated for z tending to infinity within a sector bounded away from the negative real axis, so it says nothing about behaviour near the poles.
- The recurrence at 5.5.1 is what carries the function off the positive integers; treating the factorial as the definition leaves the rest of the plane undefined.
Invariants
- Gamma(z+1) = z Gamma(z), DLMF 5.5.1
- Gamma(z) Gamma(1-z) = pi / sin(pi z), DLMF 5.5.3
- Gamma(2z) expressed through Gamma(z) and Gamma(z+1/2), DLMF 5.5.5
Reproducible procedure
- Reduce an awkward argument using the recurrence at 5.5.1.
- Move an argument across the half-plane with the reflection formula at 5.5.3, observing its excluded points.
- For large argument inside the stated sector, use the Stirling expansion at 5.11.1 and truncate before the terms begin to grow.
Error and boundary controls
- The Stirling series at 5.11.1 is a Poincare asymptotic expansion, not a convergent series. Adding terms indefinitely makes the approximation worse, so useful accuracy is bounded by the smallest term rather than by the number of terms taken.
- The digamma expansion at 5.11.2 carries the same sector restriction as 5.11.1, so neither says anything about the excluded region.
- The points excluded by 5.5.3 are poles. The function is unbounded near zero and the negative integers, so an expansion valid for large argument gives no information there.
- Reducing a large argument by repeated application of the recurrence at 5.5.1 accumulates one rounding error per step, so the reduction is not free.
What this does not establish
The chapter states identities and their conditions. It does not establish that a given library computes them to any stated accuracy, and an asymptotic expansion carries no error bound from the number of terms alone.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 5: Gamma Function · National Institute of Standards and Technology
Establishes: Numbered functional relations for the gamma and digamma functions with the conditions each requires, and the Stirling asymptotic expansions with their sector of validity.
Boundary: A reference states identities at the scope its conditions declare. The reflection formula excludes the non-positive integers, and the Stirling series is a Poincare asymptotic expansion rather than a convergent one, so taking more terms eventually makes the approximation worse.
Direct answer
- The gamma function extends the factorial off the integers. DLMF 5.5.1 states the recurrence that fixes its character: Gamma(z+1) equals z times Gamma(z), which reproduces the factorial on the positive integers and defines the function elsewhere by continuation. Two further relations do the work in practice. The reflection formula at 5.5.3, Gamma(z) times Gamma(1-z) equals pi over sin(pi z), converts an argument in one half-plane into one in the other and holds for z not a non-positive integer. The duplication formula at 5.5.5 relates Gamma(2z) to Gamma(z) and Gamma(z+1/2). Large arguments are handled by the Stirling expansion at 5.11.1, valid as z tends to infinity in the sector where the phase of z is at most pi minus delta.
Mechanism and method
- Reduce an awkward argument using the recurrence at 5.5.1.
- Move an argument across the half-plane with the reflection formula at 5.5.3, observing its excluded points.
- For large argument inside the stated sector, use the Stirling expansion at 5.11.1 and truncate before the terms begin to grow.
What is measured
- Gamma(z+1) = z Gamma(z), DLMF 5.5.1
- Gamma(z) Gamma(1-z) = pi / sin(pi z), DLMF 5.5.3
- Gamma(2z) expressed through Gamma(z) and Gamma(z+1/2), DLMF 5.5.5
Limitations
- The Stirling series at 5.11.1 is a Poincare asymptotic expansion, not a convergent series. Adding terms indefinitely makes the approximation worse, so useful accuracy is bounded by the smallest term rather than by the number of terms taken.
- The digamma expansion at 5.11.2 carries the same sector restriction as 5.11.1, so neither says anything about the excluded region.
- The points excluded by 5.5.3 are poles. The function is unbounded near zero and the negative integers, so an expansion valid for large argument gives no information there.
- Reducing a large argument by repeated application of the recurrence at 5.5.1 accumulates one rounding error per step, so the reduction is not free.
- The reflection formula at 5.5.3 requires z to be neither zero nor a negative integer.
- The duplication formula at 5.5.5 requires 2z to be neither zero nor a negative integer, which is a stricter condition than the one on 5.5.3.
- The Stirling expansion at 5.11.1 is stated for z tending to infinity within a sector bounded away from the negative real axis, so it says nothing about behaviour near the poles.
- The recurrence at 5.5.1 is what carries the function off the positive integers; treating the factorial as the definition leaves the rest of the plane undefined.
What this does not establish
- The chapter states identities and their conditions. It does not establish that a given library computes them to any stated accuracy, and an asymptotic expansion carries no error bound from the number of terms alone.