KnowraFormal power seriesLinked fromLinked fromThe 19 pages that link to Formal power series, each with the reason it gives.All 19Related 7Narrower topic 5Compared with 7Taylor seriesCompared with: A Taylor series can be manipulated formally even when it does not converge to its function.Generating functionNarrower topic: Generating functions are formal power series equipped with an interpretation for their coefficients.Power seriesCompared with: It shares the coefficient notation but does not depend on analytic convergence.Polynomial ringCompared with: Unlike polynomials, formal power series may have infinitely many nonzero coefficients.Radius of convergenceCompared with: A formal series has no convergence radius unless analytic convergence is separately considered.Asymptotic expansionCompared with: Formal manipulation alone does not establish the remainder estimates of an asymptotic expansion.Cauchy productCompared with: Formal Cauchy products are defined coefficientwise even when no analytic sums exist.Padé approximantRelated: Padé matching can be defined from coefficients alone, even when the series diverges.Exponential generating functionNarrower topic: The generating function can be defined formally, independent of analytic convergence.q-seriesNarrower topic: Many q-series identities are first interpreted algebraically, without assigning a value to q.Baker–Campbell–Hausdorff formulaRelated: The formula can be interpreted formally even when analytic convergence is not established.Newton's identitiesRelated: Generating-function derivations use formal series, so analytic convergence is unnecessary.Lagrange inversion theoremNarrower topic: The theorem applies to power series formally, so analytic convergence need not be assumed.Rogers–Ramanujan identitiesRelated: The identities can be understood coefficient by coefficient in this setting.Cauchy–Kovalevskaya theoremRelated: The coefficient recursion first yields a formal candidate, whose convergence is essential to the theorem.Jacobian conjectureRelated: At each point, an invertible linear term yields a formal inverse expansion.Pentagonal number theoremNarrower topic: The product identity is commonly interpreted coefficient by coefficient in this setting.Weierstrass preparation theoremCompared with: The theorem has formal analogues, but its stated analytic form requires convergence.Abel's binomial theoremRelated: Formal series methods use finite polynomial identities as coefficient-extraction tools.
KnowraFormal power seriesLinked fromLinked fromThe 19 pages that link to Formal power series, each with the reason it gives.All 19Related 7Narrower topic 5Compared with 7Taylor seriesCompared with: A Taylor series can be manipulated formally even when it does not converge to its function.Generating functionNarrower topic: Generating functions are formal power series equipped with an interpretation for their coefficients.Power seriesCompared with: It shares the coefficient notation but does not depend on analytic convergence.Polynomial ringCompared with: Unlike polynomials, formal power series may have infinitely many nonzero coefficients.Radius of convergenceCompared with: A formal series has no convergence radius unless analytic convergence is separately considered.Asymptotic expansionCompared with: Formal manipulation alone does not establish the remainder estimates of an asymptotic expansion.Cauchy productCompared with: Formal Cauchy products are defined coefficientwise even when no analytic sums exist.Padé approximantRelated: Padé matching can be defined from coefficients alone, even when the series diverges.Exponential generating functionNarrower topic: The generating function can be defined formally, independent of analytic convergence.q-seriesNarrower topic: Many q-series identities are first interpreted algebraically, without assigning a value to q.Baker–Campbell–Hausdorff formulaRelated: The formula can be interpreted formally even when analytic convergence is not established.Newton's identitiesRelated: Generating-function derivations use formal series, so analytic convergence is unnecessary.Lagrange inversion theoremNarrower topic: The theorem applies to power series formally, so analytic convergence need not be assumed.Rogers–Ramanujan identitiesRelated: The identities can be understood coefficient by coefficient in this setting.Cauchy–Kovalevskaya theoremRelated: The coefficient recursion first yields a formal candidate, whose convergence is essential to the theorem.Jacobian conjectureRelated: At each point, an invertible linear term yields a formal inverse expansion.Pentagonal number theoremNarrower topic: The product identity is commonly interpreted coefficient by coefficient in this setting.Weierstrass preparation theoremCompared with: The theorem has formal analogues, but its stated analytic form requires convergence.Abel's binomial theoremRelated: Formal series methods use finite polynomial identities as coefficient-extraction tools.