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Y And H Transforms
In mathematics, the transforms and transforms are complementary pairs of integral transforms involving, respectively, the Neumann function ( Bessel function of the second kind) of order and the Struve function of the same order. For a given function , the -transform of order is given by :F(k) = \int_0^\infty f(r) Y_(kr) \sqrt \, dr The inverse of above is the -transform of the same order; for a given function , the -transform of order is given by :f(r) = \int_0^\infty F(k) \mathbf_(kr) \sqrt \, dk These transforms are closely related to the Hankel transform, as both involve Bessel functions. In problems of mathematical physics and applied mathematics, the Hankel, , transforms all may appear in problems having axial symmetry. Hankel transforms are however much more commonly seen due to their connection with the 2-dimensional Fourier transform. The , transforms appear in situations with singular behaviour on the axis of symmetry (Rooney). References * ''Bateman Man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f(t) = \int_^ (Tf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neumann Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. #Spherical Bessel functions, Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Struve Function
In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation: : x^2 \frac + x \frac + \left (x^2 - \alpha^2 \right )y = \frac introduced by . The complex number α is the order of the Struve function, and is often an integer. And further defined its second-kind version \mathbf_\alpha(x) as \mathbf_\alpha(x)=\mathbf_\alpha(x)-Y_\alpha(x). The modified Struve functions are equal to , are solutions of the non-homogeneous Bessel's differential equation: : x^2 \frac + x \frac - \left (x^2 + \alpha^2 \right )y = \frac And further defined its second-kind version \mathbf_\alpha(x) as \mathbf_\alpha(x)=\mathbf_\alpha(x)-I_\alpha(x). Definitions Since this is a non-homogeneous equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the Bessel functions, and the particular solution may be chosen as the corresponding Struve fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hankel Transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor ''k'' along the ''r'' axis. The necessary coefficient of each Bessel function in the sum, as a function of the scaling factor ''k'' constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval. Definition The Hankel transform of order \nu of a function ''f''(''r'') is given by : F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrmr, where J_\nu is the Bessel function of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axial Symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. glossary of meteorology. Retrieved 2010-04-08. For example, a without trademark or other design, or a plain white tea saucer, looks the same if it is rotated by any angle about the line passing lengthwise through its center, so it is axially symmetric. Axial symmetry can also be |
Bateman Manuscript Project
The Bateman Manuscript Project was a major effort at collation and encyclopedic compilation of the mathematical theory of special functions. It resulted in the eventual publication of five important reference volumes, under the editorship of Arthur Erdélyi. Overview The theory of special functions was a core activity of the field of applied mathematics, from the middle of the nineteenth century to the advent of high-speed electronic computing. The intricate properties of spherical harmonics, elliptic functions and other staples of problem-solving in mathematical physics, astronomy and right across the physical sciences, are not easy to document completely, absent a theory explaining the inter-relationships. Mathematical tables to perform actual calculations needed to mesh with an adequate theory of how functions could be transformed into those already tabulated. Harry Bateman, a distinguished applied mathematician, undertook the somewhat quixotic task of trying to collate the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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