Carleman's Equation

	Carleman's Equation In mathematics, Carleman's equation is a Fredholm integral equation of the first kind with a logarithmic kernel. Its solution was first given by Torsten Carleman in 1922. The equation is : \int_a^b \ln, x-t, \, y(t) \, dt = f(x) The solution for ''b'' − ''a'' ≠ 4 is : y(x) = \frac \left \int_a^b \frac +\frac \int_a^b \frac \right If ''b'' − ''a'' = 4 then the equation is solvable only if the following condition is satisfied : \int_a^b \frac = 0 In this case the solution has the form : y(x) = \frac \left \int_a^b \frac +C \right where ''C'' is an arbitrary constant. For the special case ''f''(''t'') = 1 (in which case it is necessary to have ''b'' − ''a'' ≠ 4), useful in some applications, we get : y(x) = \frac \frac References * CARLEMAN, T. (1922) Uber die Abelsche Integralgleichung mit konstanten Integrationsgrenzen. Math. Z., 15, 111–120 * Gakh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fredholm Integral Equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. Equation of the first kind A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits. An inhomogeneous Fredholm equation of the first kind is written as and the problem is, given the continuous kernel function K and the function g, to find the function f. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely K(t,s)=K(ts), and the limits of integration are ±∞, then the right hand side of the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Logarithmic Kernel Logarithmic can refer to: * Logarithm, a transcendental function in mathematics * Logarithmic scale, the use of the logarithmic function to describe measurements * Logarithmic spiral, * Logarithmic growth * Logarithmic distribution, a discrete probability distribution * Natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ... {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Torsten Carleman Torsten Carleman (8 July 1892, Visseltofta, Osby Municipality – 11 January 1949, Stockholm), born Tage Gillis Torsten Carleman, was a Swedish mathematician, known for his results in classical analysis and its applications. As the director of the Mittag-Leffler Institute for more than two decades, Carleman was the most influential mathematician in Sweden. Work The dissertation of Carleman under Erik Albert Holmgren, as well as his work in the early 1920s, was devoted to singular integral equations. He developed the spectral theory of integral operators with ''Carleman kernels'', that is, kernels ''K''(''x'', ''y'') such that ''K''(''y'', ''x'') = for almost every (''x'', ''y''), and : \int , K(x, y) , ^2 dy < \infty for almost every ''x''. In the mid-1920s, Carleman developed the theory of quasi-analytic functions. He proved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fredholm Theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. The theory is named in honour of Erik Ivar Fredholm. Overview The following sections provide a casual sketch of the place of Fredholm theory in the broader context of operator theory and functional analysis. The outline presented here is broad, whereas the difficulty of formalizing this sketch is, of course, in the details. Fredholm equation of the first kind Much of Fredholm theory concerns itself with the following integral equation for ''f'' when ''g'' and ''K'' are given: :g(x)=\int_a^b K(x,y) f(y)\,dy. This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is aske ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]