hasSeriesExpansion
P90300
predicate
Indicates that a mathematical function or expression is associated with a specific series representation (such as a Taylor or power series) that expands it around a point or region.
All labels observed (9)
| Label | Occurrences |
|---|---|
| hasSeriesExpansion canonical | 4 |
| exponentialGeneratingFunction | 2 |
| hasFourierExpansion | 2 |
| seriesExpansionAtZero | 2 |
| DobinskiFormula | 1 |
| expansionInFockBasis | 1 |
| hasLaurentExpansionAt | 1 |
| hasSeriesExpansionAtZero | 1 |
| powerSeriesExpansion | 1 |
Description generation (PDg)
The one-sentence description above was generated by prompting gpt-5.1 with the predicate name and this instruction.
Instruction
Given a predicate that represents a relationship or action between entities, generate a one-sentence description explaining its meaning. # Instructions Focus on describing the relationship, not the entities themselves. # Response Format Begin the description with \' Indicates...\'
Input
Predicate: hasSeriesExpansion
Generated description
Indicates that a mathematical function or expression is associated with a specific series representation (such as a Taylor or power series) that expands it around a point or region.
Sample triples (15)
| Subject | Object |
|---|---|
| Riemann–Siegel theta function | asymptotic series in descending powers of t ⓘ |
|
Lambert W function (later named in his honor)
surface form:
Lambert W function
|
W(z) = ∑_{n=1}^{∞} (-n)^{n-1} z^n / n! via predicate surface "hasSeriesExpansionAtZero" ⓘ |
| Glauber coherent states | |α⟩ = e^{−|α|²/2} Σ_{n=0}^∞ α^n/√(n!) |n⟩ via predicate surface "expansionInFockBasis" ⓘ |
| Bell numbers | B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!} via predicate surface "DobinskiFormula" ⓘ |
| Bell numbers | \sum_{n=0}^{\infty} B_n \frac{x^n}{n!} = e^{e^x - 1} via predicate surface "exponentialGeneratingFunction" ⓘ |
| Beta function | B(x,y)=∑_{n=0}^∞ (-1)^n C(y-1,n)/(n+x) (under suitable conditions) ⓘ |
| Jacobi theta functions | Fourier series in q ⓘ |
| OEIS A002851 | log(sum_{n>=0} 2^{n(n-1)/2} x^n/n!) via predicate surface "exponentialGeneratingFunction" ⓘ |
| modular j-invariant | j(τ) = q^{-1} + 744 + 196884 q + 21493760 q^2 + … via predicate surface "hasFourierExpansion" ⓘ |
| Langevin function | L(x) = x/3 − x^3/45 + 2x^5/945 − … ⓘ |
| Siegel modular form | yes via predicate surface "hasFourierExpansion" ⓘ |
| Koebe function | k(z) = \sum_{n=1}^{\infty} n z^n via predicate surface "powerSeriesExpansion" ⓘ |
| Hurwitz zeta function | s = 1 via predicate surface "hasLaurentExpansionAt" ⓘ |
| product logarithm | W(z) = ∑_{n=1}^{∞} [(-n)^{n-1} / n!] z^n via predicate surface "seriesExpansionAtZero" ⓘ |
| product logarithm | radius of convergence e^{-1} via predicate surface "seriesExpansionAtZero" ⓘ |