Properties of Mathematical Functions & Sequences

Function properties such as its domain, period or singularities give additional information about a function or sequence that can be used directly or inform other computations. Similarly, function properties such as its sign, monotonicity or continuity classify functions and enable the use of a whole web of theorems that are true for those classes. Both kinds of function properties can be computed for any expression formed by using special functions, making it easy and efficient to write algorithms based on these high-level concepts rather than covering each individual function.

Domain and Range

FunctionDomain the domain of a function

FunctionRange the range of a function

Injectivity and Surjectivity

FunctionInjective test whether a function is injective or one-to-one

FunctionSurjective test whether a function is surjective or onto

FunctionBijective test whether a function is bijective or one-to-one and onto

Inequality Related Properties

FunctionSign the sign of a function (positive: , negative: , )

FunctionMonotonicity the monotonicity of a function (increasing: , decreasing: , )

FunctionConvexity the convexity of a function (convex: , concave: , )

Periodicity Related Properties

FunctionPeriod the periodicity of a function or sequence

Singularities and Discontinuities

FunctionDiscontinuities find the discontinuities of a function

FunctionSingularities find the singularities of a function

FunctionPoles find the poles of a meromorphic function

Continuous, Holomorphic & Meromorphic

FunctionContinuous test whether a function is continuous

FunctionAnalytic test whether a function is analytic

FunctionMeromorphic test whether a function is meromorphic

Structural Function Classes

PolynomialExpressionQ structurally test for polynomial expression

RationalExpressionQ structurally test for rational function expression