Frobenius Method
In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form z^2 u'' + p(z)z u'+ q(z) u = 0 with u' \equiv \frac and u'' \equiv \frac. in the vicinity of the regular singular point z=0. One can divide by z^2 to obtain a differential equation of the form u'' + \fracu' + \fracu = 0 which will not be solvable with regular power series methods if either or is not analytic at . The Frobenius method enables one to create a power series solution to such a differential equation, provided that ''p''(''z'') and ''q''(''z'') are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite). History Frobenius' contribution was not so much in all the possible ''forms'' of the series solutions involved (see below). These forms had all been established earlier, by Lazarus Fuchs. The ''indicial polynomial'' (see bel ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Singular Point With Indicial Roots One Half And Negative One
Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singular: Act I'', a 2018 studio album by Sabrina Carpenter *'' Singular: Act II'', a 2019 studio album by Sabrina Carpenter Mathematics * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular matrix, a matrix that is not invertible * Singular measure, a measure or probability distribution whose support has zero Lebesgue (or other) measure * Singular cardinal, an infinite cardinal number that is not a regular cardinal * Singular point of a curve, in geometry See also * Singularity (other) * Singulair Montelukast, sold under the brand name Singulair among others, is a medication used in the maintenance treatment of asthma. It is generally less preferred for this use than inhaled corticosteroids. It is ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Rational Function
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. How ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894. Definition The Laurent series for a complex function f(z) about an arbitrary point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where the coefficients a_n are defined by a contour integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic ( analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red in th ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Regular Singular Point
In mathematics, in the theory of ordinary differential equations in the complex plane \Complex, the points of \Complex are classified into ''ordinary points'', at which the equation's coefficients are analytic functions, and ''singular points'', at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different. Formal definitions More precisely, consider an ordinary linear differential equation of -th order f^(z) + \sum_^ p_i(z) f^ (z) = 0 with meromorphic functions. The equa ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Fuchs' Theorem
In mathematics, Fuchs's theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form y'' + p(x)y' + q(x)y = g(x) has a solution expressible by a generalised Frobenius series when p(x), q(x) and g(x) are analytic at x = a or a is a regular singular point. That is, any solution to this second-order differential equation can be written as y = \sum_^\infty a_n (x - a)^, \quad a_0 \neq 0 for some positive real ''s'', or y = y_0 \ln(x - a) + \sum_^\infty b_n(x - a)^, \quad b_0 \neq 0 for some positive real ''r'', where ''y''0 is a solution of the first kind. Its radius of convergence is at least as large as the minimum of the radii of convergence of p(x), q(x) and g(x). See also * Frobenius method In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form z^2 u'' + p(z)z u'+ q(z) u = 0 with u' \equiv \frac a ... ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Kummer's Equation
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term ''confluent'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. * Whittaker functions (for E ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Generalized Hypergeometric Series
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients is ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Lazarus Fuchs
Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-German mathematician who contributed important research in the field of linear differential equations. He was born in Mosina, Moschin in the Grand Duchy of Posen (modern-day Mosina, Poland) and died in Berlin, German Empire, Germany. He was buried in Schöneberg in the Alter St.-Matthäus-Kirchhof, St. Matthew's Cemetery. His grave in section H is preserved and listed as a grave of honour of the State of Berlin. Contribution He is the eponym of Fuchsian groups and functions, and the Picard–Fuchs equation. A singularity (mathematics), singular point ''a'' of a linear differential equation :y''+p(x)y'+q(x)y=0 is called Fuchsian if ''p'' and ''q'' are meromorphic function, meromorphic around the point ''a'', and have poles of orders at most 1 and 2, respectively. According to a Fuchs's theorem, theorem of Fuchs, this condition is necessary and sufficient for the regular singular point, regularity of the singular ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots in which the coefficients a_0, a_1, \dots a ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Power Series Solution Of Differential Equations
In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients. Method Consider the second-order linear differential equation a_2(z)f''(z)+a_1(z)f'(z)+a_0(z)f(z)=0. Suppose is nonzero for all . Then we can divide throughout to obtain f''+f'+f=0. Suppose further that and are analytic functions. The power series method calls for the construction of a power series solution f=\sum_^\infty A_kz^k. If is zero for some , then the Frobenius method, a variation on this method, is suited to deal with so called " singular points". The method works analogously for higher order equations as well as for systems. Example usage Let us look at the Hermite differential equation, f''-2zf'+\lambda f=0; \; \lambda=1 We can try to construct a series ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |