Series, Limits & Asymptotics

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Abstractions about convergence, asymptotic behavior, functional series, limiting operations, integral representations, and notation for multivariate analysis.

18 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Absolute convergence — Convergence of a series or improper integral after replacing every summand or integrand value by its absolute magnitude.
• Arithmetic underflow — A finite-precision arithmetic condition in which a nonzero result's magnitude is too small for normal representation in the destination format, causing a subnormal result, rounded zero or signaled exception.
• Asymptotic analysis — The study of approximations and expansions that become accurate as a variable approaches a specified limit, emphasizing dominant scales and controlled remainder behavior.
• Auxiliary function — A deliberately constructed function with engineered zeros, growth or arithmetic properties that converts a target claim—especially in transcendence theory—into an estimate or contradiction.
• Coefficient — A multiplicative factor attached to a term in an algebraic expression, series, equation or linear combination, determining that term's scale under a stated basis or representation.
• Conditional convergence — Convergence of a series or improper integral despite failure of convergence after taking absolute values.
• Dual number — An element a+b epsilon of a two-dimensional commutative algebra with nonzero nilpotent epsilon satisfying epsilon squared equals zero.
• Function series — An infinite series whose terms are functions, with its sum and inherited properties determined by a specified mode and domain of convergence.
• Hyperbolic growth — Growth following an inverse-distance-to-a-finite-singularity form, diverging as the independent variable approaches a finite critical value.
• Implicit function — A function locally or globally determined by a relation among variables even when its dependent variable is not isolated by an explicit formula.
• Improper integral — A limit-defined extension of a definite integral to an unbounded interval, an unbounded integrand or another excluded endpoint configuration.
• Interchange of limiting operations — The analysis problem of identifying conditions under which two limits, or a limit and integration, differentiation or summation, may be applied in either order with the same result.
• Mean of a function — The domain-normalized integral of a function, giving the constant value with the same total integral over a set of finite nonzero measure.
• Multi-index notation — A tuple-based notation that compresses repeated powers, partial derivatives and combinatorial coefficients in multivariable analysis into algebra resembling one-dimensional index formulas.
• Taylor series — An infinite power series whose coefficients are a function’s derivatives at one expansion point.
• Trigonometric integral — A family of special functions defined by nonelementary integrals involving sine or cosine divided by the integration variable, including the sine and cosine integrals.
• Univariate — A mathematical or statistical expression, function, distribution or analysis involving exactly one variable rather than a jointly varying tuple.
• Weight function — A function assigning unequal influence to elements in a sum, integral, average, norm or approximation.