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).
Complex planeIn mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal x-axis, called the real axis, is formed by the real numbers, and the vertical y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vectors.
Sinc functionIn mathematics, physics and engineering, the sinc function (), denoted by sinc(x), is defined as either sinc(x)=sinxx{\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}} or sinc(x)=sinπxπx,{\displaystyle \operatorname {sinc} (x)={\frac {\sin \pi x}{\pi x}},}the latter of which is sometimes referred to as the normalized sinc function.
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.
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.
Limit of a functionPoint to which functions converge in analysisAlthough the function sinxx{\displaystyle {\tfrac {\sin x}{x}}} is not defined at zero, as x becomes closer and closer to zero, sinxx{\displaystyle {\tfrac {\sin x}{x}}} becomes arbitrarily close to 1. In other words, the limit of sinxx,{\displaystyle {\tfrac {\sin x}{x}},} as x approaches zero, equals 1.
Power seriesIn mathematics, a power series (in one variable) is an infinite series of the form ∑n=0∞an(x−c)n=a0+a1(x−c)+a2(x−c)2+…{\displaystyle \sum _{n=0}^{\infty }a_{n}\left(x-c ight)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\dots } where an{\displaystyle a_{n}} represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions.
Undefined (mathematics)In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system.
Indeterminate formIn calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.
Zeros and polesIn complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity).
Neighbourhood (mathematics)In topology and mathematical analysis, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.
Open setIn general topology and mathematical analysis, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).*As an Amazon Associate I earn from qualifying purchases.