In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f = a z + b c z + d of one complex variable z. Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere and moving the sphere to a new location and orientation in space, performing stereographic projection to the plane; these transformations preserve angles, map every straight line to a line or circle, map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line, they form a group called the Möbius group, the projective linear group PGL. Together with its subgroups, it has numerous applications in physics. Möbius transformations are named in honor of August Ferdinand Möbius. Möbius transformations are defined on the extended complex plane C ^ = C ∪. Stereographic projection identifies C ^ with a sphere, called the Riemann sphere; the Möbius transformations are the bijective conformal maps from the Riemann sphere to itself, i.e. the automorphisms of the Riemann sphere as a complex manifold.
Therefore, the set of all Möbius transformations forms a group under composition. This group is called the Möbius group, is sometimes denoted Aut ; the Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds. In physics, the identity component of the Lorentz group acts on the celestial sphere in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations; this observation is taken as the starting point of twistor theory. Certain subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces; as such, Möbius transformations play an important role in the theory of Riemann surfaces. The fundamental group of every Riemann surface is a discrete subgroup of the Möbius group.
A important discrete subgroup of the Möbius group is the modular group. Möbius transformations can be more defined in spaces of dimension n>2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, orthogonal transformations and inversions; the general form of a Möbius transformation is given by f = a z + b c z + d where a, b, c, d are any complex numbers satisfying ad − bc ≠ 0. If ad = bc, the rational function defined above is a constant since f = a z + b c z + d = a c − a d − b c c = a c and is thus not considered a Möbius transformation. In case c ≠ 0, this definition is extended to the whole Riemann sphere by defining f = ∞ and f = a c. If c =
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function, holomorphic on all of D except for a discrete set of isolated points, which are poles of the function. This terminology comes from the Ancient Greek meros, meaning "part," as opposed to holos, meaning "whole." Every meromorphic function on D can be expressed as the ratio between two holomorphic functions defined on D: any pole must coincide with a zero of the denominator. Intuitively, a meromorphic function is a ratio of two well-behaved functions; such a function will still be well-behaved, except at the points where the denominator of the fraction is zero. If the denominator has a zero at z and the numerator does not the value of the function will approach infinity. From an algebraic point of view, if D is connected the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions; this is analogous to the relationship between the integers.
In the 1930s, in group theory, a meromorphic function was a function from a group G into itself that preserved the product on the group. The image of this function was called an automorphism of G. Similarly, a homomorphic function was a function between groups that preserved the product, while a homomorphism was the image of a homomorph; this terminology is now obsolete. The term endomorphism is now used for the function itself, with no special name given to the image of the function; the term meromorph is no longer used in group theory. Since the poles of a meromorphic function are isolated, there are at most countably many; the set of poles can be infinite, as exemplified by the function f = csc z = 1 sin z. By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted and the quotient f / g can be formed unless g = 0 on a connected component of D. Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers.
In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, f = z 1 / z 2 is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as holomorphic function with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two. Unlike in dimension one, in higher dimensions there do exist complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori. All rational functions, for example f = z 3 − 2 z + 10 z 5 + 3 z − 1, are meromorphic on the whole complex plane; the functions f = e z z and f = sin z 2 as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane. The function f = e 1 z is defined in the whole complex plane except for the origin, 0. However, 0 is not a pole of this function, rather an essential singularity. Thus, this function is not meromorphic in the whole complex plane.
However, it is meromorphic on C ∖. The complex logarithm function f = ln is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane while only excluding a set of isolated points; the function f = csc 1 z = 1 sin is not meromorphic in the whole plane, since the point z = 0 is an accumulation point of poles and is thus not an isolated singularity. The function f = sin 1
Providence, Rhode Island
Providence is the capital and most populous city of the U. S. is one of the oldest cities in the United States. It was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony, he named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city is situated at the mouth of the Providence River at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool and silverware industries. Today, the city of Providence is home to eight hospitals and seven institutions of higher learning which have shifted the city's economy into service industries, though it still retains some manufacturing activity; the city is the third most populous city in New England after Worcester, Massachusetts. Providence was one of the original Thirteen Colonies. Williams and his company were compelled to leave Massachusetts Bay Colony, Providence became a refuge for persecuted religious dissenters, as Williams himself had been exiled from Massachusetts.
The city was burned to the ground in March 1676 by the Narragansetts during King Philip's War, despite the good relations between Williams and the sachems with whom the United Colonies of New England were waging war. In the year, the Rhode Island legislature formally rebuked the other colonies for provoking the war. Providence residents were among the first Patriots to spill blood in the lead-up to the American Revolutionary War during the Gaspée Affair of 1772, Rhode Island was the first of the Thirteen Colonies to renounce its allegiance to the British Crown on May 4, 1776, it was the last of the Thirteen Colonies to ratify the United States Constitution on May 29, 1790, once assurances were made that a Bill of Rights would become part of the Constitution. Following the war, Providence was the country's ninth-largest city with 7,614 people; the economy shifted from maritime endeavors to manufacturing, in particular machinery, silverware and textiles. By the start of the 20th century, Providence hosted some of the largest manufacturing plants in the country, including Brown & Sharpe, Nicholson File, Gorham Manufacturing Company.
Providence residents ratified a city charter in 1831 as the population passed 17,000. The seat of city government was located in the Market House in Market Square from 1832 to 1878, the geographic and social center of the city; the city offices outgrew this building, the City Council resolved to create a permanent municipal building in 1845. The city offices moved into the Providence City Hall in 1878. During the American Civil War, local politics split over slavery as many had ties to Southern cotton and the slave trade. Despite ambivalence concerning the war, the number of military volunteers exceeded quota, the city's manufacturing proved invaluable to the Union. Providence thrived after the war, waves of immigrants brought the population from 54,595 in 1865 to 175,597 by 1900. By the early 1900s, Providence was one of the wealthiest cities in the United States. Immigrant labor powered one of the nation's largest industrial manufacturing centers. Providence was a major manufacturer of industrial products, from steam engines to precision tools to silverware and textiles.
Giant companies were based in or near Providence, such as Brown & Sharpe, the Corliss Steam Engine Company, Babcock & Wilcox, the Grinnell Corporation, the Gorham Manufacturing Company, Nicholson File, the Fruit of the Loom textile company. From 1975 until 1982, $606 million of local and national community development funds were invested throughout the city. In the 1990s, the city pushed for revitalization, realigning the north-south railroad tracks, removing the huge rail viaduct that separated downtown from the capitol building and moving the rivers to create Waterplace Park and river walks along the rivers' banks, constructing the Fleet Skating Rink and the Providence Place Mall. Despite new investment, poverty remains an entrenched problem. 27.9 percent of the city population is living below the poverty line. Recent increases in real estate values further exacerbate problems for those at marginal income levels, as Providence had the highest rise in median housing price of any city in the United States from 2004 to 2005.
The Providence city limits enclose a small geographical region with a total area of 20.5 square miles. Providence is located at the head of Narragansett Bay, with the Providence River running into the bay through the center of the city, formed by the confluence of the Moshassuck and Woonasquatucket Rivers; the Waterplace Park amphitheater and riverwalks line the river's banks through downtown. Providence is one of many cities claimed to be founded on seven hills like Rome; the more prominent hills are: Constitution Hill, College Hill, Federal Hill. The other four are: Tockwotten Hill at Fox Point, Smith Hill, Christian Hill at Hoyle Square, Weybosset Hill at the lower end of Weybosset Street, leveled in the early 1880s. Providence has 25 official neighborhoods, though these neighborhoods are grouped together and referred to
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is defined becomes divergent; the step-wise continuation technique may, come up against difficulties. These may have an topological nature, leading to inconsistencies, they may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities need not be isolated points, its investigation was a major reason for the development of sheaf cohomology. Suppose f is an analytic function defined on a non-empty open subset U of the complex plane C. If V is a larger open subset of C, containing U, F is an analytic function defined on V such that F = f ∀ z ∈ U F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with.
Analytic continuations are unique in the following sense: if V is the connected domain of two analytic functions F1 and F2 such that U is contained in V and for all z in U F1 = F2 = f,then F1 = F2on all of V. This is because F1 − F2 is an analytic function which vanishes on the open, connected domain U of f and hence must vanish on its entire domain; this follows directly from the identity theorem for holomorphic functions. A common way to define functions in complex analysis proceeds by first specifying the function on a small domain only, extending it by analytic continuation. In practice, this continuation is done by first establishing some functional equation on the small domain and using this equation to extend the domain. Examples are the gamma function; the concept of a universal cover was first developed to define a natural domain for the analytic continuation of an analytic function. The idea of finding the maximal analytic continuation of a function in turn led to the development of the idea of Riemann surfaces.
Begin with a particular analytic function f. In this case, it's given by a power series centered at z = 1: f = ∑ k = 0 ∞ k k. By the Cauchy–Hadamard theorem, its radius of convergence is 1; that is, f is defined and analytic on the open set U = which has boundary ∂ U =. Indeed, the series diverges at z = 0 ∈ ∂ U. Pretend we don't know that f = 1 / z, focus on recentering the power series at a different point a ∈ U: f = ∑ k = 0 ∞ a k k. We'll calculate the a k's and determine whether this new power series converges in an open set V, not contained in U. If so, we will have analytically continued f to the region U ∪ V, larger than U; the distance from a to ∂ U is ρ = 1 − | a − 1 | > 0. Take 0 < r < ρ. D ∪ ∂ D ⊂ U. Using Cauchy's differentiation formula to calculate the new coefficients, a k = f k! = 1 2 π i ∫ ∂ D f d ζ k + 1 = 1 2 π i ∫
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, its discovery was a significant landmark in the development of the subject; the equation is named after Erwin Schrödinger, who derived the equation in 1925, published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. In classical mechanics, Newton's second law is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force F on the system; those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation; the concept of a wave function is a fundamental postulate of quantum mechanics.
Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be unitary, must therefore be generated by the exponential of a self-adjoint operator, the quantum Hamiltonian. This derivation is explained below. In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular and subatomic systems, but macroscopic systems even the whole universe. Schrödinger's equation is central to all applications of quantum mechanics including quantum field theory which combines special relativity with quantum mechanics. Theories of quantum gravity, such as string theory do not modify Schrödinger's equation; the Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, the path integral formulation, developed chiefly by Richard Feynman.
Paul Dirac incorporated the Schrödinger equation into a single formulation. The form of the Schrödinger equation depends on the physical situation; the most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time: where i is the imaginary unit, ℏ = h 2 π is the reduced Planck constant, Ψ is the state vector of the quantum system, t is time, H ^ is the Hamiltonian operator. The position-space wave function of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector | r ⟩, it is a scalar function, expressed as Ψ = ⟨ r | Ψ ⟩. The momentum-space wave function can be defined as Ψ ~ = ⟨ p | Ψ ⟩, where | p ⟩ is the momentum eigenvector; the most famous example is the nonrelativistic Schrödinger equation for the wave function in position space Ψ of a single particle subject to a potential V, such as that due to an electric field. Where m is the particle's mass, ∇ 2 is the Laplacian.
This is a diffusion equation, but unlike the heat equation, this one is a wave equation given the imaginary unit present in the transient term. The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version; the general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is a classical approximation to reality and yields accurate results in many situations, but only to a certain extent. To apply the Schrödinger equation, write down the Hamiltonian for the system, accounting for the kinetic and potential energies of the particles constituting the system insert it into the Schrödinger equation; the resulting partial differential equation is solved for the wave function, which contains information about the system. The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states.
These states are important as their individual study simplifies the task of solving the time-dependent Schrödinger equation for any state. Stationary states can be described by a simpler form of the Schrödinger equation, the time-independe