Mathematical Analysis

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18 domain-specific abstractions whose origin domain is Mathematical Analysis.

• Absolute convergence — Convergence of a series or improper integral after replacing every summand or integrand value by its absolute magnitude.
• 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.
• Conditional convergence — Convergence of a series or improper integral despite failure of convergence after taking absolute values.
• Function series — An infinite series whose terms are functions, with its sum and inherited properties determined by a specified mode and domain of convergence.
• Heaviside step function — A threshold function equal to zero below an origin and one above it, with a declared convention at the discontinuity.
• 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.
• Inverse Trigonometric Functions — Restrict each periodic trigonometric function to an injective branch and invert it, returning a principal angle while retaining rules for other periodic or complex branches.
• Korn's inequality — A rigidity inequality bounding the full gradient of a vector field, modulo rigid motions, by its symmetric gradient under specified domain and boundary conditions.
• Local boundedness — A property requiring a function, family or operator to remain bounded on some neighborhood of every point, without requiring one global bound over the whole domain.
• Logarithmic mean — Average two positive numbers by their difference divided by the difference of their logarithms, using the continuous value x when the arguments coincide.
• Operational Calculus — A family of methods that represents differentiation, integration, or related operators in an algebraic domain, solves the resulting operator equation, and interprets the inverse representation as a function.
• Pseudo-differential operator — An operator defined through a position- and frequency-dependent symbol, extending differential operators to non-polynomial multipliers and supporting microlocal analysis of PDE.
• Quantum calculus — Calculus built from finite q-ratio or h-shift difference operators and corresponding sums, recovering ordinary differential and integral calculus as q→1 or h→0 rather than using limits internally.
• Staircase paradox — A sequence of rectilinear curves can converge uniformly to a diagonal while their lengths fail to converge to the diagonal’s length.
• Stone–Weierstrass Theorem — A density criterion stating that a real subalgebra of continuous functions on a compact Hausdorff space uniformly approximates every continuous function when it contains constants and separates points, with conjugation closure required in the complex case.
• Weight function — A function assigning unequal influence to elements in a sum, integral, average, norm or approximation.