Working definition
DLMF 13.2.1 gives Kummer equation, with a regular singularity at the origin carrying indices zero and one minus b, and an irregular singularity of rank one at infinity. That is the same structural description the Bessel chapter gives at 10.2.1, which is what it means to call these confluent. The first solution at 13.2.2 is a series converging for all complex argument, entire in the argument and in a, and meromorphic in b, but the chapter states plainly that it does not exist when b is a non-positive integer. Olver form at 13.2.3 divides by a gamma function instead and is entire in all three parameters, with 13.2.4 relating the two by a factor of the gamma function at b. The second solution at 13.2.6 has a branch point at the origin and behaves like z to the minus a at infinity in a stated sector, and 13.2.7 and 13.2.8 give the parameter values at which the solution becomes a polynomial.
Notation
M of a, b, z for the first solution and U of a, b, z for the secondbold M for Olver entire forma and b for the parameters and z for the argumentAssumptions
- The first solution does not exist when b is a non-positive integer, which is the exception 13.2.3 exists to remove.
- The relation at 13.2.4 between the two forms holds except at those same non-positive integers.
- The asymptotic statement for the second solution at 13.2.6 holds in a stated sector rather than everywhere.
Invariants
- Kummer equation with a regular singularity at the origin and an irregular one of rank one at infinity, DLMF 13.2.1
- The first solution converges for all complex argument but fails to exist at non-positive integer b, DLMF 13.2.2
- Olver form is entire in all three parameters, DLMF 13.2.3
Reproducible procedure
- Check b against zero and the negative integers before using the first solution, and take the entire form at 13.2.3 where it fails.
- Read the second solution as branch-dependent, since 13.2.6 declares a principal branch through the principal value of z to the minus a.
- Where a parameter makes the solution a polynomial by 13.2.7 or 13.2.8, use that form rather than summing a series that has already terminated.
Error and boundary controls
- Convergence for all argument is not accuracy at all argument. The series at 13.2.2 converges everywhere and still loses significance to cancellation for large argument, which is why the second solution carries an asymptotic description instead.
- Near a non-positive integer b the first solution is ill conditioned even where it is defined, which is the practical reason for the entire form at 13.2.3, exactly as the regularized form exists in the Gauss case at 15.2.2.
- The chapter states solutions and their structure and makes no accuracy claim about any implementation.
What this does not establish
The chapter defines the solutions, their exceptional parameters and their branch structure. It does not establish that any implementation is accurate, and everywhere-convergence is not everywhere-usable.
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 13: Confluent Hypergeometric Functions · National Institute of Standards and Technology
Establishes: Kummer equation with its singular structure, the first solution as an everywhere-convergent series with the parameter values at which it fails to exist, the entire alternative form that removes them, the second solution with its branch point and asymptotic behaviour, and the parameter values at which the series terminates.
Boundary: The chapter defines the solutions and their analytic character. Convergence for all argument is not accuracy at all argument, and the chapter makes no claim about any evaluation scheme.
Direct answer
- DLMF 13.2.1 gives Kummer equation, with a regular singularity at the origin carrying indices zero and one minus b, and an irregular singularity of rank one at infinity. That is the same structural description the Bessel chapter gives at 10.2.1, which is what it means to call these confluent. The first solution at 13.2.2 is a series converging for all complex argument, entire in the argument and in a, and meromorphic in b, but the chapter states plainly that it does not exist when b is a non-positive integer. Olver form at 13.2.3 divides by a gamma function instead and is entire in all three parameters, with 13.2.4 relating the two by a factor of the gamma function at b. The second solution at 13.2.6 has a branch point at the origin and behaves like z to the minus a at infinity in a stated sector, and 13.2.7 and 13.2.8 give the parameter values at which the solution becomes a polynomial.
Mechanism and method
- Check b against zero and the negative integers before using the first solution, and take the entire form at 13.2.3 where it fails.
- Read the second solution as branch-dependent, since 13.2.6 declares a principal branch through the principal value of z to the minus a.
- Where a parameter makes the solution a polynomial by 13.2.7 or 13.2.8, use that form rather than summing a series that has already terminated.
What is measured
- Kummer equation with a regular singularity at the origin and an irregular one of rank one at infinity, DLMF 13.2.1
- The first solution converges for all complex argument but fails to exist at non-positive integer b, DLMF 13.2.2
- Olver form is entire in all three parameters, DLMF 13.2.3
Limitations
- Convergence for all argument is not accuracy at all argument. The series at 13.2.2 converges everywhere and still loses significance to cancellation for large argument, which is why the second solution carries an asymptotic description instead.
- Near a non-positive integer b the first solution is ill conditioned even where it is defined, which is the practical reason for the entire form at 13.2.3, exactly as the regularized form exists in the Gauss case at 15.2.2.
- The chapter states solutions and their structure and makes no accuracy claim about any implementation.
- The first solution does not exist when b is a non-positive integer, which is the exception 13.2.3 exists to remove.
- The relation at 13.2.4 between the two forms holds except at those same non-positive integers.
- The asymptotic statement for the second solution at 13.2.6 holds in a stated sector rather than everywhere.
What this does not establish
- The chapter defines the solutions, their exceptional parameters and their branch structure. It does not establish that any implementation is accurate, and everywhere-convergence is not everywhere-usable.