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Elliptic Rational Function
In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name). Rational elliptic functions are identified by a positive integer order ''n'' and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree ''n'' in ''x'' with selectivity factor ΞΎ is generally defined as: :R_n(\xi,x)\equiv \mathrm\left(n\frac\,\mathrm^(x,1/\xi),1/L_n(\xi)\right) where * cd(u,k) is the Jacobi elliptic cosine function. * K() is a complete elliptic integral of the first kind. * L_n(\xi)=R_n(\xi,\xi) is the discrimination factor, equal to the minimum value of the magnitude of R_n(\xi,x) for , x, \ge\xi. For many cases, in particular for orders of the form ''n'' = 2'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Elliptic Functions (n=1,2,3,4, X=--1,1-)
Rationality is the Quality (philosophy), quality of being guided by or based on reason. In this regard, a person Action (philosophy), acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ability, as in a rational animal, to a psychological process, like Logical reasoning, reasoning, to mental states, such as beliefs and intentions, or to persons who possess these other forms of rationality. A thing that lacks rationality is either ''arational'', if it is outside the domain of rational evaluation, or ''irrational'', if it belongs to this domain but does not fulfill its standards. There are many discussions about the Essence, essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella, which is why it is rational for an agent to do so in re ... [...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] |
Rational Functions
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. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a filter (signal processing), signal processing filter with equalized ripple (filters), ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in Gain (electronics), gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations. As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Rational Functions
In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree is defined as: :R_n(x)\ \stackrel\ T_n\left(\frac\right) where is a Chebyshev polynomial of the first kind. Properties Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves. Recursion :R_(x)=2\left(\frac\right)R_(x)-R_(x)\quad\text\,n\ge 1 Differential equations :(x+1)^2R_n(x)=\frac\fracR_(x)-\frac\fracR_(x) \quad \text n\ge 2 :(x+1)^2x\fracR_n(x)+\frac\fracR_n(x)+n^2R_(x) = 0 Orthogonality Defining: :\omega(x) \ \stackrel\ \frac The orthogonality of the Chebyshev rational functions may be written: :\int_^\infty R_m(x)\,R_n(x)\,\omega(x)\,\mathrmx=\frac\delta_ where for and for ; is the Kronecker delta In mathematics, the Kronecker delta (named after Leop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi's Elliptic Functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacobi elliptic functions denoted by \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legendre canonical forms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Polynomials
The Chebyshev polynomials are two sequences of orthogonal 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 is not obvious at first sight but can be shown using de Moivre's formula (see below). The Chebyshev polynomials are polynomials with the largest possible leading coefficient whose absolute value on the interval is bounded by 1. They are also the "extremal" polynomials for many other properties. In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems; the roots of , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Elliptic Function (abs, N=3, X=(0,5))
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ability, as in a rational animal, to a psychological process, like reasoning, to mental states, such as beliefs and intentions, or to persons who possess these other forms of rationality. A thing that lacks rationality is either ''arational'', if it is outside the domain of rational evaluation, or ''irrational'', if it belongs to this domain but does not fulfill its standards. There are many discussions about the essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella, which is why it is rational for an agent to do so in response. An important rival to this approach are coherence-based accoun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Elliptic Functions (xi Varied, X=(-1,1))
Rationality is the Quality (philosophy), quality of being guided by or based on reason. In this regard, a person Action (philosophy), acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ability, as in a rational animal, to a psychological process, like Logical reasoning, reasoning, to mental states, such as beliefs and intentions, or to persons who possess these other forms of rationality. A thing that lacks rationality is either ''arational'', if it is outside the domain of rational evaluation, or ''irrational'', if it belongs to this domain but does not fulfill its standards. There are many discussions about the Essence, essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella, which is why it is rational for an agent to do so in re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Functions
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. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
