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Essential Singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non-isolated singularities too; those do not have a residue. Formal description Consider an open subset U of the complex plane \mathbb. Let a be an element of U, and f\colon U\setminus\\to \mathbb a holomorphic function. The point a is called an ''essential singularity'' of the function f if the singularity is neither a pole nor a removable singularity. For example, the function f(z)=e^ has an essential singularity at z=0. Alternative descriptions Let a be a complex number, and assume that f(z) is not defined at a but i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Neighbourhood (mathematics)
In topology and related areas of mathematics, 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 (topology), interior. Intuitively speaking, a neighbourhood of a point is a Set (mathematics), set of points containing that point where one can move some amount in any direction away from that point without leaving the set. Definitions Neighbourhood of a point If X is a topological space and p is a point in X, then a neighbourhood of p is a subset V of X that includes an open set U containing p, p \in U \subseteq V \subseteq X. This is equivalent to the point p \in X belonging to the Interior (topology)#Interior point, topological interior of V in X. The neighbourhood V need not be an open subset of X. When V is open (resp. closed, compact, etc.) in X, it is called an (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors require neighbourhoods to be open, so i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stephen Wolfram
Stephen Wolfram ( ; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer algebra and theoretical physics. In 2012, he was named a fellow of the American Mathematical Society. As a businessman, he is the founder and CEO of the software company Wolfram Research, where he works as chief designer of Mathematica and the Wolfram Alpha answer engine. Early life Family Stephen Wolfram was born in London in 1959 to Hugo and Sybil Wolfram, both German Jewish refugees to the United Kingdom. His maternal grandmother was British psychoanalyst Kate Friedlander. Wolfram's father, Hugo Wolfram, was a textile manufacturer and served as managing director of the Lurex Company—makers of the fabric Lurex. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Picard's Great Theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by \lambda, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If f omits two values, then the composition of f with the inverse of the modular function maps the plane into the unit disc which implies that f is constant by Liouville's theorem. This theorem is a significant strengthening of Liouville's theorem which states that the image ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Casorati–Weierstrass Theorem
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem, because it was discovered independently by Sokhotski in 1868. Formal statement of the theorem Start with some open subset U in the complex plane containing the number z_0, and a function f that is holomorphic on U \setminus \, but has an essential singularity at z_0 . The ''Casorati–Weierstrass theorem'' then states that This can also be stated as follows: Or in still more descriptive terms: The theorem is considerably strengthened by Picard's great theorem, which states, in the notation above, that f assumes ''every'' complex value, with one possible exception, infinitely often on V. In the case that f is an entire function and a = \infty, the theorem says that the values f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Meromorphic
In the mathematical field of complex analysis, a meromorphic function on an open set, open subset ''D'' of the complex plane is a function (mathematics), function that is holomorphic function, holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), ''poles'' of the function. The term comes from the Greek language, Greek ''meros'' (wikt:μέρος, μέρος), meaning "part". Every meromorphic function on ''D'' can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator. Heuristic description Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will approach infinity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Point At Infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point). In the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the Pole (complex analysis), poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Principal Part
In mathematics, the principal part has several independent meanings but usually refers to the negative-power portion of the Laurent series of a function. Laurent series definition The principal part at z=a of a function : f(z) = \sum_^\infty a_k (z-a)^k is the portion of the Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ... consisting of terms with negative degree. That is, : \sum_^\infty a_ (z-a)^ is the principal part of f at a . If the Laurent series has an inner radius of convergence of 0, then f(z) has an essential singularity at a if and only if the principal part is an infinite sum. If the inner radius of convergence is not 0, then f(z) may be regular at a despite the Laurent series having an infinite principal part. Other definitions Calculus Consider the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894. Definition The Laurent series for a complex function f(z) about an arbitrary point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where the coefficients a_n are defined by a contour integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic ( analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
