Working definition
Bessel equation is stated at DLMF 10.2.1 as z squared times the second derivative, plus z times the first derivative, plus the quantity z squared minus nu squared times the function, equal to zero. The chapter records its analytic structure directly: a regular singularity at the origin with indices plus and minus nu, and an irregular singularity of rank one at infinity. The first-kind solution at 10.2.2 is a power series whose coefficients divide by the gamma function at nu plus k plus one, which is why the gamma function turns up in a problem that began as a differential equation. The second-kind solution at 10.2.3 is built from the first at plus and minus nu, and 10.2.4 gives the limiting form when nu is an integer and that construction degenerates.
Notation
J of nu at z for the first kindY of nu at z for the second kindnu for the order and z for the argumentAssumptions
- The first-kind solution at 10.2.2 is analytic except for a branch point at the origin when nu is not an integer.
- The second-kind solution has a branch point at the origin whether or not nu is an integer, with a cut along the negative real axis.
- The construction at 10.2.3 divides by sine of nu pi, so it degenerates at integer order and 10.2.4 supplies the limit instead.
- The equation has a regular singularity at zero and an irregular one at infinity, so behaviour at the two ends is not governed by the same expansion.
Invariants
- Bessel equation, DLMF 10.2.1
- The first-kind series with its gamma-function coefficients, DLMF 10.2.2
- The second-kind solution built from first-kind solutions of opposite order, DLMF 10.2.3
Reproducible procedure
- Fix the order nu, and note whether it is an integer, because that decides whether 10.2.3 or 10.2.4 applies.
- Choose a solution pair from Table 10.2.1 for the region in question rather than assuming one pair works everywhere.
- Respect the branch cut along the negative real axis when continuing either solution.
- Where the order is large or the argument small, check that the chosen pair is still the numerically satisfactory one for that region.
Error and boundary controls
- Linear independence and numerical satisfactoriness are different properties. A pair can be independent and still lose all its accuracy to cancellation, which is why Table 10.2.1 is organised by region.
- The series at 10.2.2 alternates, so for large argument it subtracts nearly equal terms and loses significance long before it stops converging.
- The construction at 10.2.3 divides by sine of nu pi, so near integer order it is ill conditioned even where it is defined.
What this does not establish
The chapter gives the solutions and their analytic structure. It does not select a pair for a given computation, and it makes no claim that any particular implementation is accurate over any particular range.
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 10: Bessel Functions · National Institute of Standards and Technology
Establishes: Bessel equation with its singularity structure, the series definition of the first-kind solution, the construction of the second-kind solution, and which solution pairs are numerically satisfactory in which region.
Boundary: A reference states the solutions and their analytic structure. It does not choose a pair for a particular computation: linear independence and numerical satisfactoriness are different properties, and the reference tabulates the second by region rather than asserting one pair is always right.
Direct answer
- Bessel equation is stated at DLMF 10.2.1 as z squared times the second derivative, plus z times the first derivative, plus the quantity z squared minus nu squared times the function, equal to zero. The chapter records its analytic structure directly: a regular singularity at the origin with indices plus and minus nu, and an irregular singularity of rank one at infinity. The first-kind solution at 10.2.2 is a power series whose coefficients divide by the gamma function at nu plus k plus one, which is why the gamma function turns up in a problem that began as a differential equation. The second-kind solution at 10.2.3 is built from the first at plus and minus nu, and 10.2.4 gives the limiting form when nu is an integer and that construction degenerates.
Mechanism and method
- Fix the order nu, and note whether it is an integer, because that decides whether 10.2.3 or 10.2.4 applies.
- Choose a solution pair from Table 10.2.1 for the region in question rather than assuming one pair works everywhere.
- Respect the branch cut along the negative real axis when continuing either solution.
- Where the order is large or the argument small, check that the chosen pair is still the numerically satisfactory one for that region.
What is measured
- Bessel equation, DLMF 10.2.1
- The first-kind series with its gamma-function coefficients, DLMF 10.2.2
- The second-kind solution built from first-kind solutions of opposite order, DLMF 10.2.3
Limitations
- Linear independence and numerical satisfactoriness are different properties. A pair can be independent and still lose all its accuracy to cancellation, which is why Table 10.2.1 is organised by region.
- The series at 10.2.2 alternates, so for large argument it subtracts nearly equal terms and loses significance long before it stops converging.
- The construction at 10.2.3 divides by sine of nu pi, so near integer order it is ill conditioned even where it is defined.
- The first-kind solution at 10.2.2 is analytic except for a branch point at the origin when nu is not an integer.
- The second-kind solution has a branch point at the origin whether or not nu is an integer, with a cut along the negative real axis.
- The construction at 10.2.3 divides by sine of nu pi, so it degenerates at integer order and 10.2.4 supplies the limit instead.
- The equation has a regular singularity at zero and an irregular one at infinity, so behaviour at the two ends is not governed by the same expansion.
What this does not establish
- The chapter gives the solutions and their analytic structure. It does not select a pair for a given computation, and it makes no claim that any particular implementation is accurate over any particular range.