Elliptic function

In complex analysis, an elliptic function is a meromorphic function, periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which repeat in a lattice; such a doubly periodic function cannot be holomorphic, as it would be a bounded entire function, by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of all simple poles must cancel. Elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, their theory was improved by Carl Gustav Jacobi. Jacobi's elliptic functions have found numerous applications in physics, were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed.

Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms. Formally, an elliptic function is a function f meromorphic on ℂ for which there exist two non-zero complex numbers ω1 and ω2 with ω1/ω2 ∉ ℝ, such that f = f and f = f for all z ∈ ℂ. Denoting the "lattice of periods" by Λ =, it follows that f = f for all ω ∈ Λ. There are those of Weierstrass. Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors follow Weierstrass when presenting the elementary theory, because his functions are simpler, any elliptic function can be expressed in terms of them. With the definition of elliptic functions given above the Weierstrass elliptic function ℘ is constructed in the most obvious way: given a lattice Λ as above, put ℘ = 1 z 2 + ∑ ω ∈ Λ ∖ This function is invariant with respect to the transformation z ↦ z + ω for any ω ∈ Λ.

The addition of the −1/ω2 terms is necessary to make the sum converge. The technical condition to ensure that an infinite sum such as this converges to a meromorphic function is that on any compact set, after omitting the finitely many terms having poles in that set, the remaining series converges normally. On any compact disk defined by |z| ≤ R, for any |ω| > 2R, one has | 1 2 − 1 ω 2 | = | 2 ω z − z 2 ω 2 2 | = | z ω 3 2 | ≤ 10 R | ω | 3 and it can be shown that the sum ∑ ω ≠ 0 1 | ω | 3 converges regardless of Λ. By writing ℘ as a Laurent series and explicitly comparing terms, one may verify that it satisfies the relation 2 = 4 3 − g 2 ℘ − g 3 where g 2 = 60 ∑ ω ∈ Λ ∖ 1 ω 4 {\displaystyle g_=60\sum _{\omeg

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous. A smooth function is a function. Differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer; the function f is said to be of class Ck if the derivatives f′, f′′... F are continuous; the function f is said to be of class C ∞, or smooth. The function f is said to be of class Cω, or analytic, if f is smooth and if its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. Cω is thus contained in C∞. Bump functions are examples of functions in C∞ but not in Cω. To put it differently, the class C0 consists of all continuous functions; the class C1 consists of all differentiable functions.

Thus, a C1 function is a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k, there are examples to show that this containment is strict. C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers; the function f = { x if x ≥ 0, 0 if x < 0 is continuous, but not differentiable at x = 0, so it is of class C0 but not of class C1. The function g = { x 2 sin if x ≠ 0, 0 if x = 0 is differentiable, with derivative g ′ = { − cos + 2 x sin if x ≠ 0, 0 if x = 0; because cos oscillates as x → 0, g’ is not continuous at zero. Therefore, g is differentiable but not of class C1. Moreover, if one takes g = x4/3sin in this example, it can be used to show that the derivative function of a differentiable function can be unbounded on a compact set and, that a differentiable function on a compact set may not be locally Lipschitz continuous.

The functions f = | x | k + 1 where k is are continuous and k times differentiable at all x. But at x = 0 they are not times differentiable, so they are of class Ck but not of class Cj where j > k. The exponential function is analytic, so, of class Cω; the trigonometric functions are analytic wherever they are defined. The function f = { e − 1 1 − x 2 if | x | < 1, 0 otherwise is smooth, so of class C∞, but it is not analytic at x = ±1, so it is not of class Cω. The function f is an example of a smooth function with compact support. A function f: U ⊂ R n → R defined on an open set U of R n is said to be of class C k {\displayst

Curve

In mathematics, a curve is speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line. Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition. A curve is a topological space, locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is its ending point—that is, a path from any of its points to the same point. Related meanings include the graph of a function and a two-dimensional graph. Interest in curves began; this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times.

Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach. The term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length", while a straight line is defined as "a line that lies evenly with the points on itself". Euclid's idea of a line is clarified by the statement "The extremities of a line are points,". Commentators further classified lines according to various schemes. For example: Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction; these curves include: The conic sections studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles and used as a method to double the cube.

The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle; the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century; this enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, those that cannot, transcendental curves. Curves had been described as "geometrical" or "mechanical" according to how they were, or could be, generated. Conic sections were applied in astronomy by Kepler. Newton worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways.

The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into'ovals'; the statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. Since the nineteenth century there has not been a separate theory of curves, but rather the appearance of curves as the one-dimensional aspect of projective geometry, differential geometry; the era of the space-filling curves provoked the modern definitions of curve. In general, a curve is defined through a continuous function γ: I → X from an interval I of the real numbers into a topological space X. Depending on the context, it is either γ or its image γ, called a curve. In general topology, when non-differentiable functions are considered, it is the map γ, called a curve, because its image may look differently from what is called a curve.

For example, the image of the Peano curve fills the square. On the other hand, when one considers curves defined by a differentiable function, this is the image of the function, called a curve; the curve is said to be simple, or a Jordan arc, if γ is injective, i.e. if for all x, y in I, we have γ = γ

Cusp (singularity)

In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic parametric equation x = f y = g, a cusp is a point where both derivatives of f and g are zero, the directional derivative, in the direction of the tangent, changes sign. Cusps are local singularities in the sense that they involve only one value of the parameter t, in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by smooth implicit equation F = 0, cusps are points where the terms of lowest degree of the Taylor expansion of F are a power of a linear polynomial; the theory of Puiseux series implies that, if F is an analytic function, a linear change of coordinates allows the curve to be parametrized, in a neighborhood of the cusp, as x = a t m y = S, where a is a real number, m is a positive integer, S is a power series of order k larger than m.

The number m is sometimes called the order or the multiplicity of the cusp, is equal to the degree of the nonzero part of lowest degree of F. These definitions have been generalized to curves defined by differentiable functions by René Thom and Vladimir Arnold, in the following way. A curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps. In some contexts, in the remainder of this article, the definition of a cusp is restricted to the case of cusps of order two—that is, the case where m = 2. A plane curve cusp may be put in the following form by a diffeomorphism of the plane: x2 – y2k+1 = 0, where k is a positive integer. Consider a smooth real-valued function of two variables, say f where x and y are real numbers. So f is a function from the plane to the line; the space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target.

This action splits the whole function space up into equivalence classes, i.e. orbits of the group action. One such family of equivalence classes is denoted by Ak ±; this notation was introduced by V. I. Arnold. A function f is said to be of type Ak± if it lies in the orbit of x2 ± yk+1, i.e. there exists a diffeomorphic change of coordinate in source and target which takes f into one of these forms. These simple forms x2 ± yk+1 are said to give normal forms for the type Ak±-singularities. Notice that the A2n+ are the same as the A2n− since the diffeomorphic change of coordinate → in the source takes x2 + y2n+1 to x2 − y2n+1. So we can drop the ± from A2n± notation; the cusps are given by the zero-level-sets of the representatives of the A2n equivalence classes, where n ≥ 1 is an integer. An ordinary cusp is given by x2 − y3 i.e. the zero-level-set of a type A2-singularity. Let f be a smooth function of x and y and assume, for simplicity, that f = 0. A type A2-singularity of f at can be characterised by:Having a degenerate quadratic part, i.e. the quadratic terms in the Taylor series of f form a perfect square, say L2, where L is linear in x and y, L does not divide the cubic terms in the Taylor series of f.

A rhamphoid cusp denoted a cusp such that both branches are on the same side of the tangent, such as for the curve of equation x 2 − x 4 − x 5 = 0. As such a singularity is in the same differential class as the cusp of equation x 2 − x 5 = 0, a singularity of type A4, the term has been extended to all such singularities; these cusps are non-generic as wavefronts. The rhamphoid cusp and the ordinary cusp are non-diffeomorphic. For a type A4-singularity we need f to have a degenerate quadratic part, that L does divide the cubic terms, another divisibility condition, a final non-divisibility condition. To see where these extra divisibility conditions come from, assume that f has a degenerate quadratic part L2 and that L divides the cubic terms, it follows that the third order

Conic section

In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, the ellipse; the circle is a special case of the ellipse, is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties; the conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, some particular line, called a directrix, are in a fixed ratio, called the eccentricity; the type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.

This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically; the conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone, it shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines; these are called degenerate conics and some authors do not consider them to be conics at all.

Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics, the ellipse and hyperbola; the circle is a special kind of ellipse, although it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the plane is a closed curve; the circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to one generating line of the cone the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L.

For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, for e > 1 a hyperbola. A circle is not defined by a focus and directrix, in the plane; the eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane, its directrix. An ellipse and a hyperbola each have distinct directrices for each of them; the line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis called transverse axis in the hyperbola; the midpoint of this line segment is called the center of the conic. Let a denote the distance from the center to a vertex of an ellipse or hyperbola; the distance from the center to a directrix is a/e while the distance from the center to a focus is ae. A parabola does not have a center; the eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.

If the angle between the surface of the cone and its axis is β and the angle between the cutting plane and the axis is α, the eccentricity is cos α cos β. A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, the center in these cases is the midpoint of the line segment joining the foci; some of the other common features and/or. The linear eccentricity is the distance between the focus; the latus rectum is the chord parallel to the directrix and passing through the focus. Its length is denoted by 2ℓ; the semi-latus rectum is half of the length of the latus rec

Mathematics

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

Cramer's paradox

In mathematics, Cramer's paradox or the Cramer–Euler paradox is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitrary points that are needed to define one such curve. It is named after the Genevan mathematician Gabriel Cramer; this paradox is the result of a naive understanding or a misapplication of two theorems: Bézout's theorem. Cramer's theorem. Observe that for all n ≥ 3, n2 ≥ n/2, so it would naively appear that for degree three or higher there could be enough points shared by each of two curves that those points should determine either of the curves uniquely; the resolution of the paradox is that in certain degenerate cases n / 2 points are not enough to determine a curve uniquely. The paradox was first published by Colin Maclaurin. Cramer and Leonhard Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer, it has become known as Cramer's paradox after featuring in his 1750 book Introduction à l'analyse des lignes courbes algébriques, although Cramer quoted Maclaurin as the source of the statement.

At about the same time, Euler published examples showing a cubic curve, not uniquely defined by 9 points and discussed the problem in his book Introductio in analysin infinitorum. The result was explained by Julius Plücker. For first order curves the paradox does not occur, because n = 1 so n2 = 1 < n / 2 = 2. In general two distinct lines L1 and L2 intersect at a single point P unless the lines are of equal gradient, in which case they do not intersect at all. A single point is not sufficient to define a line. Two nondegenerate conics intersect at most at 4 finite points in the real plane, fewer than the 32 = 9 given as a maximum by Bézout's theorem, 5 points are needed to define a nondegenerate conic. In a letter to Euler, Cramer pointed out that the cubic curves x3 − x = 0 and y3 − y = 0 intersect in 9 points. Hence 9 points are not sufficient to uniquely determine a cubic curve in degenerate cases such as these. A bivariate equation of degree n has 1 + n / 2 coefficients, but the set of points described by the equation is preserved if the equation is divided through by one of the coefficients, leaving one coefficient equal to 1 and only n / 2 coefficients to characterize the curve.

Given n / 2 points, each of these points can be used to create a separate equation by substituting it into the general polynomial equation of degree n, giving n / 2 equations linear in the n / 2 unknown coefficients. If this system is non-degenerate in the sense of having a non-zero determinant, the unknown coefficients are uniquely determined and hence the polynomial equation and its curve are uniquely determined, but if this determinant is zero, the system is degenerate and the points can be on more than one curve of degree n. Ed Sandifer "Cramer’s Paradox" Cramer's Paradox at MathPages