Differentiable functionIn mathematical analysis, a real or complexfunction of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable, the graph of a differentiable function has a non-verticaltangent line at each interior point in its domain. A differentiable function is locally approximable by a linear function at each interior point, and does not contain any break, angle, or cusp.Continuous functionIn mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting its argument to sufficiently small changes. A discontinuous function is a function that is not continuous.
Holomorphic functionIn mathematics, a holomorphic function is a complex-valued function of one or morecomplex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate spaceCn{\displaystyle \mathbb {C} ^{n}}. The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic).
Analytic functionIn mathematical analysis, an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at a point if, in some neighborhood of that point, it is equal to a power series centered there. The analytic function is therefore locally determined by coefficients of the series, or equivalently by derivatives of the function evaluated at the center of the series expansion.
Partial differential equationIn mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations.
Differentiable manifoldIn mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply.
Function (mathematics)In mathematics, a function from a setX to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
Differential topologyIn mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape.
DerivativeIn mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value.
Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex-valued functions of one or more complex variables. It is used in many branches of mathematics, including functional analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory.
Mathematical analysisMathematical analysis is the branch of mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It grew out of calculus, especially the use of derivatives and integrals to study variable quantities, and in the 19th century its foundations were reformulated with greater rigor. Basic objects of study in mathematical analysis include the real numbers, functions, sequences, series, and limits.
Absolute valueIn mathematics, the absolute value or modulus of a real numberx{\displaystyle x}, denoted |x|{\displaystyle |x|}, is the (non-negative) magnitudeof x{\displaystyle x}measured without regard to its sign. Namely, |x|=x{\displaystyle |x|=x} if x{\displaystyle x} is a positive number, and |x|=−x{\displaystyle |x|=-x} if x{\displaystyle x} is negative (in which case −x{\displaystyle -x} is positive), and |0|=0{\displaystyle |0|=0}.*As an Amazon Associate I earn from qualifying purchases.