Working definition
DLMF 15.2.1 defines the Gauss hypergeometric function as a series in Pochhammer symbols over the unit disk, extended elsewhere by analytic continuation, with the principal branch taken in the sector where the phase of one minus z is at most pi and a cut running from one to infinity along the real axis. The chapter states plainly that the function does not in general exist when the lower parameter c is zero or a negative integer. The regularized form at 15.2.2 divides by a gamma function instead and is valid for all c, which is why it exists. 15.2.4 records that the series terminates into a polynomial when the upper parameter is a non-positive integer.
Notation
F of a, b; c; z for the Gauss functionbold F for the regularized formthe Pochhammer symbol for the rising factorialAssumptions
- The series at 15.2.1 converges on the open unit disk; values outside come from continuation, not summation.
- The function is generally undefined when c is zero or a negative integer, which is the exception the regularized form at 15.2.2 removes.
- A principal branch is declared with a cut from one to infinity, so a value is branch-dependent rather than absolute.
Invariants
- The Gauss series on the unit disk, DLMF 15.2.1
- The regularized form valid for all c, DLMF 15.2.2
- Termination into a polynomial at non-positive integer upper parameter, DLMF 15.2.4
Reproducible procedure
- Check c against zero and the negative integers before using the unregularized form.
- On the boundary circle, classify by the real part of c minus a minus b: above zero converges absolutely, between minus one and zero converges conditionally away from z equals one, and at or below minus one diverges.
- Where a parameter makes the series terminate, use the polynomial form rather than summing an infinite series that has already stopped.
Error and boundary controls
- The three boundary regimes are stated by condition rather than pointwise, so conditional convergence near the circle is slow and the chapter offers no rate.
- Continuation outside the unit disk is an analytic statement, not a numerical method; the chapter does not bound the accuracy of any particular continuation scheme.
- Near c equal to a non-positive integer the unregularized form is ill conditioned even where it is defined, which is the practical reason for 15.2.2.
What this does not establish
The chapter defines the function, its branch and its exceptional parameters. It does not establish the accuracy of any continuation scheme outside the unit disk.
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 15: Hypergeometric Function · National Institute of Standards and Technology
Establishes: The Gauss series with its unit disk of convergence, the parameter values for which it is undefined, the regularized form that removes those exceptions, the termination condition, and the three convergence regimes on the boundary circle.
Boundary: A definition with a declared principal branch and cut. It does not establish that a given numerical continuation outside the disk is accurate, and the boundary behaviour is stated by regime rather than pointwise.
Direct answer
- DLMF 15.2.1 defines the Gauss hypergeometric function as a series in Pochhammer symbols over the unit disk, extended elsewhere by analytic continuation, with the principal branch taken in the sector where the phase of one minus z is at most pi and a cut running from one to infinity along the real axis. The chapter states plainly that the function does not in general exist when the lower parameter c is zero or a negative integer. The regularized form at 15.2.2 divides by a gamma function instead and is valid for all c, which is why it exists. 15.2.4 records that the series terminates into a polynomial when the upper parameter is a non-positive integer.
Mechanism and method
- Check c against zero and the negative integers before using the unregularized form.
- On the boundary circle, classify by the real part of c minus a minus b: above zero converges absolutely, between minus one and zero converges conditionally away from z equals one, and at or below minus one diverges.
- Where a parameter makes the series terminate, use the polynomial form rather than summing an infinite series that has already stopped.
What is measured
- The Gauss series on the unit disk, DLMF 15.2.1
- The regularized form valid for all c, DLMF 15.2.2
- Termination into a polynomial at non-positive integer upper parameter, DLMF 15.2.4
Limitations
- The three boundary regimes are stated by condition rather than pointwise, so conditional convergence near the circle is slow and the chapter offers no rate.
- Continuation outside the unit disk is an analytic statement, not a numerical method; the chapter does not bound the accuracy of any particular continuation scheme.
- Near c equal to a non-positive integer the unregularized form is ill conditioned even where it is defined, which is the practical reason for 15.2.2.
- The series at 15.2.1 converges on the open unit disk; values outside come from continuation, not summation.
- The function is generally undefined when c is zero or a negative integer, which is the exception the regularized form at 15.2.2 removes.
- A principal branch is declared with a cut from one to infinity, so a value is branch-dependent rather than absolute.
What this does not establish
- The chapter defines the function, its branch and its exceptional parameters. It does not establish the accuracy of any continuation scheme outside the unit disk.