# Calculus The 24 definitions of the calculus library, each with its Epsil spelling, its MathJSON name, its signature and its full description. Each definition is listed under its Epsil spelling (the MathJSON name when it has none), with its signature in the engine's type syntax. The [Standard Library](/library/) page is the one-page index of every category. ## Definitions ### bigO MathJSON `BigO` · `(value) -> number` Landau big-O remainder term. Inert; any numeric approximation (.N()) of an expression containing it is NaN. ### circleContour MathJSON `CircleContour` · `(center: complex, radius: real, orientation: integer?) -> expression` Closed circle: center, positive radius, optional orientation (+1 or -1). ### circularIntegrate MathJSON `CircularIntegrate` · `(function, limits+) -> number` Closed-path integral. Evaluates supported explicit contours by the residue theorem. ### contourIntegrate MathJSON `ContourIntegrate` · `(expression, variable: symbol, contour: expression) -> number` Symbolic integral over an explicit closed contour, using the residue theorem. ### D `(expression, variables: symbol*) -> expression` Symbolic partial derivative with respect to one or more variables. ### dSolve MathJSON `DSolve` · `(expression, symbol, symbol) -> expression` Symbolic differential equation solver. ### derivative MathJSON `Derivative` · `(function, order: number*) -> function` Derivative operator that returns a derivative function. ### integrate MathJSON `Integrate` · `(function, limits+) -> list | list | number | tuple` Symbolic integral with optional bounds. ### interpolatingFunction MathJSON `InterpolatingFunction` · `(list, number?) -> number` Piecewise-quartic dense-output interpolant of a numeric ODE solution (produced by `NDSolveFunction`). The first operand is the per-step coefficient table; applied to a number, it evaluates the solution there (clamping to the covered interval outside it). Stays symbolic for a non-numeric argument. ### jacobianMatrix MathJSON `JacobianMatrix` · `(any, any?) -> value` JacobianMatrix(fs, vars): the matrix of partial derivatives ∂fᵢ/∂xⱼ, one row per function and one column per variable. `fs` is a list of expressions. A single (non-list) expression is the gradient case: the result is the flat vector [∂f/∂x₁, …, ∂f/∂xₙ]. `vars` is a list of symbols and may be omitted, in which case the free variables of `fs` are used, in lexicographic order. Example: JacobianMatrix([x^2 y, x + z], [x, y, z]). ### limit MathJSON `Limit` · `(function, point: number, direction: number?) -> number` Limit of a function ### Limits `(index: symbol, lower: value, upper: value) -> tuple` Limits of a function ### nd MathJSON `ND` · `(function, at: number) -> list | number | tuple` Numerical derivative evaluated at a point. ### ndSolve MathJSON `NDSolve` · `(expression, symbol, limits: symbol | tuple, number, number?) -> list` Numerical differential equation solver. ### ndSolveFunction MathJSON `NDSolveFunction` · `(expression, symbol, limits: symbol | tuple, number) -> function` Numerically solve an ordinary differential equation and return the solution as an applicable function (a `Function` literal wrapping an `InterpolatingFunction`), usable at any point of the integration interval. Same arguments as `NDSolve`, without the sample count. ### nIntegrate MathJSON `NIntegrate` · `(function, lower: number, upper: number) -> number` Numerical approximation of a definite integral. ### nLimit MathJSON `NLimit` · `(function, point: number, direction: number?) -> number` Numerical approximation of the limit of a function ### normal MathJSON `Normal` · `(value) -> value` Strip Big-O remainder terms from a series, yielding the truncated polynomial. Example: Normal(Series(\sin x, x)) → x - x^3/6 + x^5/120 ### polygonContour MathJSON `PolygonContour` · `(vertices: list, orientation: integer?) -> expression` Simple closed polygon: a list of complex vertices in traversal order, with optional orientation override (+1 or -1). ### rSolve MathJSON `RSolve` · `(expression, symbol, symbol) -> expression` Symbolic recurrence equation solver. ### realLineContour MathJSON `RealLineContour` · `(principalValue: boolean?) -> expression` The real axis from minus infinity to infinity. Pass True to explicitly request a Cauchy principal value. ### rectangleContour MathJSON `RectangleContour` · `(lowerLeft: complex, upperRight: complex, orientation: integer?) -> expression` Closed rectangle: lower-left and upper-right complex corners, optional orientation (+1 or -1). ### residue MathJSON `Residue` · `(expression, variable: symbol, point: value) -> number` Residue of a function at a point (the coefficient of (x-a)⁻¹ in its Laurent expansion) ### series MathJSON `Series` · `(expression, variable: symbol?, point: value?, order: number?) -> number` Taylor series expansion of an expression about a point (or an asymptotic expansion at ±∞), including Laurent, Puiseux (fractional-power), and log-aware expansions at poles and branch points. Only essential singularities, irrational exponents, and nested/reciprocal logarithms are left unevaluated. Example: Series(\sin x, x) → x - x^3/6 + x^5/120 + O(x^7)