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Trigonometric Series
In mathematics, trigonometric series are a special class of orthogonal series of the form : A_0 + \sum_^\infty A_n \cos + B_n \sin, where x is the variable and \ and \ are coefficients. It is an infinite version of a trigonometric polynomial. A trigonometric series is called the Fourier series of the integrable function f if the coefficients have the form: :A_n=\frac1\pi \int^_0\! f(x) \cos \,dx :B_n=\frac\displaystyle\int^_0\! f(x) \sin \, dx Examples Every Fourier series gives an example of a trigonometric series. Let the function f(x) = x on \pi,\pi/math> be extended periodically (see sawtooth wave). Then its Fourier coefficients are: :\begin A_n &= \frac1\pi\int_^ x \cos \,dx = 0, \quad n \ge 0. \\ ptB_n &= \frac1\pi\int_^ x \sin \, dx \\ pt&= -\frac \cos + \frac1\sin \Bigg\vert_^\pi \\ mu&= \frac, \quad n \ge 1.\end Which gives an example of a trigonometric series: :2\sum_^\infty \frac \sin = 2\sin - \frac22\sin + \frac23\sin - \frac24\sin + \cdots However ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality (mathematics)
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity'' to linear algebra of bilinear forms. Two elements and of a vector space with bilinear form B are orthogonal when B(\mathbf,\mathbf)= 0. Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces, families of functions are used to form an orthogonal basis (linear algebra), basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics. Definitions * In geometry, two Euclidean vectors are orthogonal if they are perpendicular, ''i.e.'' they form a right angle. * Two vector space, vectors and in an inner product space V are ''orthogonal'' if their inner product \langle \mathbf, \mathbf \rangle is zero. This relationship is denoted \mathbf \perp \mathbf. * A set of vectors in an inner produ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series (mathematics)
In mathematics, a series is, roughly speaking, an addition of Infinity, infinitely many Addition#Terms, terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greece, Ancient Greeks, the idea that a potential infinity, potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the Quadrature of the Parabola, quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless quantity, number without units, in which case it is known as a numerical factor. It may also be a constant (mathematics), constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any mathematical expression, expression (including Variable (mathematics), variables such as , and ). When the combination of variables and constants is not necessarily involved in a product (mathematics), product, it may be called a ''parameter''. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. A , also known as constant term or simply constant, is a quantity either implicitly attach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform. The term ''trigonometric polynomial'' for the real-valued case can be seen as using the analogy: the functions sin(''nx'') and cos(''nx'') are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of e^, i.e., Laurent polynomials in z under the change of variables x \mapsto z := e^. Def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable Function
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an ''antiderivative'', a function whose derivat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series For The Identity Function
Fourier may refer to: * Fourier (surname), French surname Mathematics *Fourier series, a weighted sum of sinusoids having a common period, the result of Fourier analysis of a periodic function *Fourier analysis, the description of functions as sums of sinusoids *Fourier transform, the type of linear canonical transform that is the generalization of the Fourier series *Fourier operator, the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform *Fourier inversion theorem, any one of several theorems by which Fourier inversion recovers a function from its Fourier transform *Short-time Fourier transform or short-term Fourier transform (STFT), a Fourier transform during a short term of time, used in the area of signal analysis *Fractional Fourier transform (FRFT), a linear transformation generalizing the Fourier transform, used in the area of harmonic analysis *Discrete-time Fourier transform (DTFT), the reverse of the Fourier series, a special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sawtooth Wave
The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform. The convention is that a sawtooth wave ramps upward and then sharply drops. In a reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the extreme case of an asymmetric triangle wave. The equivalent piecewise linear functions x(t) = t - \lfloor t \rfloor x(t) = t \bmod 1 based on the floor function of time ''t'' is an example of a sawtooth wave with period 1. A more general form, in the range −1 to 1, and with period ''p'', is 2\left( - \left\lfloor + \right\rfloor\right) This sawtooth function has the same phase as the sine function. While a square wave is constructed from only odd harmonics, a sawtooth wave's sound is harsh and clear and its spectrum cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Series With Wiggles And Spike
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle measur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite set, infinite and well-order, well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal number, cardinal and ordinal number, ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Wey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Set (mathematics)
In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S. It is usually denoted by S'. The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line. Definition The derived set of a subset S of a topological space X, denoted by S', is the set of all points x \in X that are limit points of S, that is, points x such that every neighbourhood of x contains a point of S other than x itself. Examples If \Reals is endowed with its usual Euclidean topology then the derived set of the half-open interval , 1) is the closed interval [0, 1 Consider \Reals with the Topology (structure)">topology (open sets) consisting of the empty set and any subset of \Reals that contains 1. The derived set of A := \ is A' = \Reals \setminus \. Properties Let X denote a topological space in what follows. If A and B are subsets of X, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
