Chebyshev Equation

	Chebyshev Equation Chebyshev's equation is the second order linear differential equation : (1-x^2) - x + p^2 y = 0 where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions can be obtained by power series: :y = \sum_^\infty a_nx^n where the coefficients obey the recurrence relation : a_ = a_n. The series converges for , x, <1 (note, ''x'' may be complex), as may be seen by applying the ratio test to the recurrence. The recurrence may be started with arbitrary values of ''a''₀ and ''a''₁, leading to the two-dimensional space of solutions that arises from second order differential equations. The standard choices are: :''a''₀ = 1 ; ''a''₁ = 0, leading to the solution : $F(x) = 1 - \fracx^2 + \fracx^4 - \fracx^6 + \cdots$ and :''a''₀ = 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ordinary Differential Equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Russia Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the world, with its internationally recognised territory covering , and encompassing one-eighth of Earth's inhabitable landmass. Russia extends across Time in Russia, eleven time zones and shares Borders of Russia, land boundaries with fourteen countries, more than List of countries and territories by land borders, any other country but China. It is the List of countries and dependencies by population, world's ninth-most populous country and List of European countries by population, Europe's most populous country, with a population of 146 million people. The country's capital and List of cities and towns in Russia by population, largest city is Moscow, the List of European cities by population within city limits, largest city entirely within E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias. Transcription The surname Chebyshev has been transliterated in several different ways, like Tchebichef, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Čebyčev, Čebyšev, Chebysheff, Chebychov, Chebyshov (according to native Russian speakers, this one provides the closest pronunciation in English to the correct pronunciation in old Russian), and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Power Series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Recurrence Relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Ratio Test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The test The usual form of the test makes use of the limit The ratio test states that: * if ''L'' 1 then the series diverges; * if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case. It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let :R = \lim\sup \left, \frac\ :r = \lim\inf \left, \frac\. Then the ratio test states that: * if ''R'' 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Chebyshev Polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind T_n are defined by : T_n(\cos \theta) = \cos(n\theta). Similarly, the Chebyshev polynomials of the second kind U_n are defined by : U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big). That these expressions define polynomials in \cos\theta may not be obvious at first sight, but follows by rewriting \cos(n\theta) and \sin\big((n+1)\theta\big) using de Moivre's formula or by using the angle sum formulas for \cos and \sin repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain T_2(\cos\theta)=\cos(2\theta)=2\cos^2\theta-1 and U_1(\cos\theta)\sin\theta=\sin(2\theta)=2\cos\theta\sin\theta, which are respectively a polynomial in \cos\th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]