Smooth function in the context of "Hyperbola"

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions.

A function is analytic if and only if for every ${\displaystyle x_{0}}$ in its domain, its Taylor series about ${\displaystyle x_{0}}$ converges to the function in some neighborhood of ${\displaystyle x_{0}}$. This is stronger than merely being infinitely differentiable at ${\displaystyle x_{0}}$, and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic.

In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation ⁠${\displaystyle \log(xy)=\log(x)+\log(y)}$⁠.

If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have highly irregular solutions. For example, the gamma function is a function that satisfies the functional equation ${\displaystyle f(x+1)=xf(x)}$ and the initial value ${\displaystyle f(1)=1.}$ There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem).

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.

In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f." (This is by Sard's theorem.)

There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions of "almost none" (negligible set); the two main classes are:

In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.

In mathematical analysis, a bump function (also called a test function) is a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ on a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain ${\displaystyle \mathbb {R} ^{n}}$ forms a vector space, denoted ${\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}$ or ${\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}$ The dual space of this space endowed with a suitable topology is the space of distributions.

In calculus, Taylor's theorem gives an approximation of a ${\textstyle k}$-times differentiable function around a given point by a polynomial of degree ${\textstyle k}$, called the ${\textstyle k}$-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ${\textstyle k}$ of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.

Taylor's theorem is named after Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory.