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Fuchs Relation
In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called ''Fuchsian equations''. It is named after Lazarus Immanuel Fuchs. Definition Fuchsian equation A linear differential equation in which every singular point, including the point at infinity, is a regular singularity is called ''Fuchsian equation'' or ''equation of Fuchsian type''. For Fuchsian equations a formal fundamental system exists at any point, due to the Fuchsian theory. Coefficients of a Fuchsian equation Let a_1, \dots, a_r \in \mathbb be the r regular singularities in the finite part of the complex plane of the linear differential equationLf := \frac + q_1\frac + \cdots + q_\frac + q_nf with meromorphic functions q_i. For linear differential equations the singularities are exactly the singular points of the coefficients. Lf=0 is a Fuchsian equation if and only if the coefficients are rational functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Linear Differential Equation
In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. Types of solution A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of antiderivative, integrals. This is also true for a linear equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularities
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the reciprocal function f(x) = 1/x has a singularity at x = 0, where the value of the function is not defined, as involving a division by zero. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: typ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Fuchsian Theory
The Fuchsian theory of linear differential equation In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) wher ...s, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them. At any Regular singular point, ordinary point of a homogeneous linear differential equation of order n there exists a fundamental system of n linearly independent power series solutions. A non-ordinary point is called a singularity. At a Regular singular point, singularity the maximal number of linearly independent power series solutions may be less than the order of the differential equation. Generalized series solutions The generalized series at \xi\in\mathbb is defined by : (z-\xi)^\alpha\sum_^\infty c_k(z-\xi)^k, \text \alpha,c_k \in \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meromorphic Function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. The term comes from the Greek ''meros'' ( μέρος), meaning "part". Every meromorphic function on ''D'' can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator. Heuristic description Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at ''z'', then one must compare the multiplicity of these zeros. From an algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Fuchsian Theory
The Fuchsian theory of linear differential equation In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) wher ...s, which is named after Lazarus Immanuel Fuchs, provides a characterization of various types of singularities and the relations among them. At any Regular singular point, ordinary point of a homogeneous linear differential equation of order n there exists a fundamental system of n linearly independent power series solutions. A non-ordinary point is called a singularity. At a Regular singular point, singularity the maximal number of linearly independent power series solutions may be less than the order of the differential equation. Generalized series solutions The generalized series at \xi\in\mathbb is defined by : (z-\xi)^\alpha\sum_^\infty c_k(z-\xi)^k, \text \alpha,c_k \in \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
