Branch pointIn the mathematical field of complex analysis, a branch point of a multivalued function is a point such that if the function is n{\displaystyle n}-valued (has n{\displaystyle n} values) at that point, all of its neighborhoods contain a point that has more than n{\displaystyle n} values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.
Exponential functionMathematical function, denoted exp(x) or e^xIn mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted ex{\displaystyle e^{x}} or expx{\displaystyle \exp x}; the latter is preferred when the argument x{\displaystyle x} is a complicated expression.
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.
Natural logarithmLogarithm to the base of the mathematical constant eThe natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x).
Complex planeThe complex plane. The horizontal axis represents the real numbers(R){\displaystyle (\mathbb {R} )}, the vertical axis represents the imaginary numbers(I){\displaystyle (\mathbb {I} )}.
Covering spaceType of continuous map in topologyIntuitively, a covering locally projects a "stack of pancakes" above an open neighborhoodU{\displaystyle U} onto U{\displaystyle U}In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms.
Analytic continuationIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation that initially defined the function becomes divergent. The step-wise continuation technique may, however, come up against difficulties.
Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
Inverse functionIn mathematics, the inverse function of a functionf (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only iff is bijective, and if it exists, is denoted by f−1.{\displaystyle f^{-1}.}For a function f:X→Y{\displaystyle f\colon X\to Y}, its inverse f−1:Y→X{\displaystyle f^{-1}\colon Y\to X} admits an explicit description: it sends each element y∈Y{\displaystyle y\in Y} to the unique element x∈X{\displaystyle x\in X} such that f(x) = y. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7.
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.
Continuous functionMathematical function with no sudden changes*As an Amazon Associate I earn from qualifying purchases.