Global analytic function in the context of Elementary function

Global analytic function in the context of Elementary function

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⭐ Core Definition: Global analytic function

In the mathematical field of complex analysis, a global analytic function (or complete analytic function) is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analytic functions arise naturally in considering the possible analytic continuations of an analytic function, since analytic continuations may have a non-trivial monodromy. They are one foundation for the theory of Riemann surfaces.

The definition of a global analytic function goes back to Karl Weierstrass.

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👉 Global analytic function in the context of Elementary function

In mathematics, an elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions, as well as those functions obtained by addition, multiplication, division, and composition of these. Some functions which are encountered by beginners are not elementary, such as the absolute value function and piecewise-defined functions. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all algebraic functions (not often encountered by beginners), and all functions obtained by roots of a polynomial whose coefficients are elementary.

This list of elementary functions was originally set forth by Joseph Liouville in 1833. A key property is that all elementary functions have derivatives of any order, which are also elementary, and can be algorithmically computed by applying the differentiation rules (or the rules for implicit differentiation in the case of roots). The Taylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they are global analytic functions, defined (possibly with multiple values, such as the elementary function or ) for every complex argument, except at isolated points. In contrast, antiderivatives of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.

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Global analytic function in the context of Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds. An example of a Riemannian manifold is a surface, on which distances are measured by the length of curves on the surface. Riemannian geometry is the study of surfaces and their higher-dimensional analogs (called manifolds), in which distances are calculated along curves belonging to the manifold. Formally, Riemannian geometry is the study of smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

View the full Wikipedia page for Riemannian geometry
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← Riemannian geometry ← Elementary function
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Mathematics → Complex analysis → Analytic function → Branch cut → Analytic continuation → Monodromy → Riemann surface → Karl Weierstrass →