Measurable functionIn mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spacespreserves the topological structure: the preimage of any open set is open.
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}.Function (mathematics)In mathematics, a function from a set X 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.
Domain of a functionIn mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom(f){\displaystyle \operatorname {dom} (f)} or domf{\displaystyle \operatorname {dom} f}, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".
Lebesgue integralMethod of mathematical integration The integral of a positive function can be interpreted as the area under a curve. In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Complex numberNumber with a real and an imaginary partA complex number z = a + bi can be visually represented as a pair of numbers (a, b) forming a position vector (blue) or a point (red) on an Argand diagram, representing the complex plane.
Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
Absolute convergenceMode of convergence of an infinite seriesIn mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.Henstock–Kurzweil integralIn mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced), Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of inequivalent definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral.
Improper integralConcept in mathematical analysis An improper Riemann integral of the first kind, where the region in the plane implied by the integral is infinite in extent horizontally. The area of such a region, which the integral represents, may be finite (as here) or infinite. An improper Riemann integral of the second kind, where the implied region is infinite vertically. The region may have either finite (as here) or infinite area.
Integration by substitutionTechnique in integral evaluation In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards." This involves differential forms.*As an Amazon Associate I earn from qualifying purchases.