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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 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<number> | list<tuple> | number | tuple

Symbolic integral with optional bounds.

interpolatingFunction​

MathJSON InterpolatingFunction · (list<any>, 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> | 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<complex>, 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)