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Fourier–Bessel Series
In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series (an infinite series expansion on a finite interval) based on Bessel functions. Fourier–Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems. Definition The Fourier–Bessel series of a function with a domain of satisfying f: ,b\to \R is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind ''J''''α'', where the argument to each version ''n'' is differently scaled, according to (J_\alpha )_n (x) := J_\alpha \left( \fracb x \right) where ''u''''α'',''n'' is a root, numbered ''n'' associated with the Bessel function ''J''''α'' and ''c''''n'' are the assigned coefficients: f(x) \sim \sum_^\infty c_n J_\alpha \left( \fracb x \right). Interpretation The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlömilch's Series
Schlömilch's series is a Fourier series type expansion of twice continuously differentiable function in the interval (0,\pi) in terms of the Bessel function of the first kind, named after the German mathematician Oskar Schlömilch, who derived the series in 1857. The real-valued function f(x) has the following expansion: :f(x) = a_0 + \sum_^\infty a_n J_0(nx), where :\begin a_0 &= f(0) + \frac \int_0^\pi \int_0^ u f'(u\sin\theta)\ d\theta\ du, \\ a_n &= \frac \int_0^\pi \int_0^ u\cos nu \ f'(u\sin\theta)\ d\theta\ du. \end Examples Some examples of Schlömilch's series are the following: *Null functions in the interval (0,\pi) can be expressed by Schlömilch's Series, 0 = \frac+\sum_^\infty (-1)^n J_0(nx), which cannot be obtained by Fourier Series. This is particularly interesting because the null function is represented by a series expansion in which not all the coefficients are zero. The series converges only when 0 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neumann Polynomial
In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions. The first few polynomials are :O_0^(t)=\frac 1 t, :O_1^(t)=2\frac , :O_2^(t)=\frac + 4\frac , :O_3^(t)=2\frac + 8\frac , :O_4^(t)=\frac + 4\frac + 16\frac . A general form for the polynomial is :O_n^(t)= \frac \sum_^ (-1)^\frac \left(\frac 2 t \right)^, and they have the "generating function" :\frac \frac 1 = \sum_O_n^(t) J_(z), where ''J'' are Bessel functions. To expand a function ''f'' in the form :f(z)=\sum_ a_n J_(z)\, for , z, |
Kapteyn Series
Kapteyn series is a series expansion of analytic functions on a domain in terms of the Bessel function of the first kind. Kapteyn series are named after Willem Kapteyn, who first studied such series in 1893.Kapteyn, W. (1893). Recherches sur les functions de Fourier-Bessel. Ann. Sci. de l’École Norm. Sup., 3, 91-120. Let f be a function analytic on the domain :D_a = \left\ with a0, \Theta_n(z) is defined by : \Theta_n(z) = \frac14\sum_^\frac\left(\frac\right)^ Kapteyn's series are important in physical problems. Among other applications, the solution E of Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was first derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his '' Epi ... M=E-e\sin E can be expressed via a Kapteyn series: : E=M+2\sum_^\infty\fracJ_n(ne). Relation between the Taylor coefficients and the \alpha_n coefficients ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Fourier Series
In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we consider that of square-integrable functions defined on an interval of the real line, which is important, among others, for interpolation theory. Definition Consider a set of square-integrable functions with values in \mathbb = \Complex or \mathbb = \R, \Phi = \_^\infty, which are pairwise orthogonal for the inner product \langle f, g\rangle_w = \int_a^b f(x)\,\overline(x)\,w(x)\,dx where w(x) is a weight function, and \overline\cdot represents complex conjugation, i.e., \overline(x) = g(x) for \mathbb = \R. The generalized Fourier series of a square-integrable function f : , b\to \mathbb, with respect to Φ, is then f(x) \sim \sum_^\infty c_n\varphi_n(x), where the coefficients are given by c_n = . If Φ is a complete set, i.e., an orthogonal basis of the space of all square-in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robin Boundary Condition
In mathematics, the Robin boundary condition (; properly ), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary differential equation, ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function (mathematics), function ''and'' the values of its derivative on the boundary (topology), boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition. Definition Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts to mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also called impedance boundary conditions, from their application in Electromagnetism, electromagnetic problems, or convective boundary conditio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Projection
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straight line parallel to . It is a vector parallel to , defined as: \mathbf_1 = a_1\mathbf where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . In turn, the scalar projection is defined as: a_1 = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and ''θ'' is the angle between and . Which finally gives: \mathbf_1 = \left(\mathbf \cdot \mathbf\right) \mathbf = \frac \frac = \frac = \frac ~ . The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . The vector component or vector resolute of perpendi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two vectors in the space is a Scalar (mathematics), scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite Dimension (vector space), dimension are widely used in functional analysis. Inner product spaces over the Field (mathematics), field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
