Working definition
DLMF 2.1.1 defines asymptotic equality by the ratio: f is asymptotically equal to phi when the ratio of the two tends to one in the limit. The order symbols follow, with little-o at 2.1.2 when the ratio tends to zero and big-O at 2.1.3 when the ratio stays bounded. A Poincare asymptotic expansion is defined at 2.1.13 and 2.1.14 by a condition on every truncation: for each n the function equals the first n terms plus a remainder of order x to the minus n. The chapter states that a convergent series is automatically the asymptotic expansion of its sum, but that the converse fails, and it records that the functions zero, e to the minus z, and e to the minus z cosine z all share one null expansion in suitable sectors.
Notation
The tilde for asymptotic equalitybig-O for a bounded ratio and little-o for a vanishing onex tending to c within a declared point setAssumptions
- Every one of these statements is relative to a limit point and a point set, so an order symbol without a stated limit says nothing.
- The expansion condition at 2.1.13 is imposed on each truncation separately rather than on the infinite series.
- A convergent series is an asymptotic expansion of its sum, but an asymptotic expansion need not converge.
Invariants
- Asymptotic equality is the ratio tending to one, DLMF 2.1.1
- Big-O is a bounded ratio and little-o a vanishing one, DLMF 2.1.2 and 2.1.3
- A Poincare expansion is a condition on every truncation, DLMF 2.1.13 and 2.1.14
Reproducible procedure
- State the limit point and the point set before writing an order symbol, since the symbol is meaningless without them.
- Read an expansion as a family of statements about truncations rather than as a series to be summed.
- Where an expansion is used numerically, truncate at the smallest term rather than at as many terms as are available.
Error and boundary controls
- An asymptotic expansion does not determine its function. The chapter gives zero, e to the minus z, and e to the minus z cosine z as three functions sharing one null expansion, so agreement of expansions is not agreement of functions.
- The defining condition bounds the remainder for each fixed truncation as the argument goes to the limit. It says nothing about accuracy at a fixed argument as more terms are taken.
- A divergent expansion has a smallest term, and accuracy past that point degrades however carefully the arithmetic is done.
What this does not establish
The chapter defines what asymptotic statements mean. It does not establish accuracy at any particular argument, and it states explicitly that an expansion is shared by more than one function.
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 2: Asymptotic Approximations · National Institute of Standards and Technology
Establishes: Definitions for asymptotic equality and the order symbols, the definition of a Poincare asymptotic expansion, and the statement that such an expansion does not determine the function it describes.
Boundary: The chapter defines what an asymptotic statement means. It carries no claim that any particular expansion is accurate at a particular argument, and it records explicitly that distinct functions can share one expansion.
Direct answer
- DLMF 2.1.1 defines asymptotic equality by the ratio: f is asymptotically equal to phi when the ratio of the two tends to one in the limit. The order symbols follow, with little-o at 2.1.2 when the ratio tends to zero and big-O at 2.1.3 when the ratio stays bounded. A Poincare asymptotic expansion is defined at 2.1.13 and 2.1.14 by a condition on every truncation: for each n the function equals the first n terms plus a remainder of order x to the minus n. The chapter states that a convergent series is automatically the asymptotic expansion of its sum, but that the converse fails, and it records that the functions zero, e to the minus z, and e to the minus z cosine z all share one null expansion in suitable sectors.
Mechanism and method
- State the limit point and the point set before writing an order symbol, since the symbol is meaningless without them.
- Read an expansion as a family of statements about truncations rather than as a series to be summed.
- Where an expansion is used numerically, truncate at the smallest term rather than at as many terms as are available.
What is measured
- Asymptotic equality is the ratio tending to one, DLMF 2.1.1
- Big-O is a bounded ratio and little-o a vanishing one, DLMF 2.1.2 and 2.1.3
- A Poincare expansion is a condition on every truncation, DLMF 2.1.13 and 2.1.14
Limitations
- An asymptotic expansion does not determine its function. The chapter gives zero, e to the minus z, and e to the minus z cosine z as three functions sharing one null expansion, so agreement of expansions is not agreement of functions.
- The defining condition bounds the remainder for each fixed truncation as the argument goes to the limit. It says nothing about accuracy at a fixed argument as more terms are taken.
- A divergent expansion has a smallest term, and accuracy past that point degrades however carefully the arithmetic is done.
- Every one of these statements is relative to a limit point and a point set, so an order symbol without a stated limit says nothing.
- The expansion condition at 2.1.13 is imposed on each truncation separately rather than on the infinite series.
- A convergent series is an asymptotic expansion of its sum, but an asymptotic expansion need not converge.
What this does not establish
- The chapter defines what asymptotic statements mean. It does not establish accuracy at any particular argument, and it states explicitly that an expansion is shared by more than one function.