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List Of Things Named After Friedrich Bessel
{{Short description, none This is a (partial) list of things named for Friedrich Wilhelm Bessel, a 19th-century German scholar who worked in astronomy, geodesy and mathematical sciences: Astronomy, geodesy, astronomical bodies * 1552 Bessel * Bessel's star; see 61 Cygni * Bessel (crater) * Bessel ellipsoid *Besselian elements * Besselian epoch * Bessel points * Repsoid–Bessel pendulum Mathematics * Bessel's correction * Bessel's differential equation * Bessel's inequality * Bessel potential * Bessel potential spaces * Bessel process Bessel and related functions * Bessel beam * Bessel filter * Bessel function ** Bessel–Maitland function ** Incomplete Bessel functions * Bessel polynomial **q-Bessel polynomials *Bessel series, see Fourier–Bessel series * Bessel window *Bessel–Clifford function *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 Bes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrich Wilhelm Bessel
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method of parallax. A special type of mathematical functions were named Bessel functions after Bessel's death, though they had originally been discovered by Daniel Bernoulli and then generalised by Bessel. Life and family Bessel was born in Minden, Westphalia, then capital of the Prussian administrative region Minden-Ravensberg, as second son of a civil servant into a large family. At the age of 14 Bessel was apprenticed to the import-export concern Kulenkamp at Bremen. The business's reliance on cargo ships led him to turn his mathematical skills to problems in navigation. This in turn led to an interest in astronomy as a way of determining longitude. Bessel came to the attention of a major figure of German astronomy at the time, Heinrich Wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are many c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel–Clifford Function
In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If :\pi(x) = \frac = \frac is the entire function defined by means of the reciprocal gamma function, then the Bessel–Clifford function is defined by the series :_n(z) = \sum_^\infty \pi(k+n) \frac The ratio of successive terms is ''z''/''k''(''n'' + ''k''), which for all values of ''z'' and ''n'' tends to zero with increasing ''k''. By the ratio test, this series converges absolutely for all ''z'' and ''n'', and uniformly for all regions with bounded , ''z'', , and hence the Bessel–Clifford function is an entire function of the two complex variables ''n'' and ''z''. Differential equation of the Bessel–Clifford function It follows from the above series on differentiating with respect to ''x'' that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Window
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Q-Bessel Polynomials
In mathematics, the ''q''-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called hy ...s by Roelof Koekoek, Peter Lesky Rene Swarttouw, Hypergeometric Orthogonal Polynomials and their q-Analogues, p526 Springer 2010： :y_(x;a;q)=\;_2\phi_1 \left(\begin q^ & -aq^ \\ 0 \end ; q,qx \right). Also known as alternative q-Charlier polynomials K(x;a;q). Orthogonality : \sum_^\left(\frac*q^*y_*(q^k;a;q)*y_*(q^k;a;q)\right)=(q;q)_*(-aq^n;q)_\frac\delta_ Roelof p527 where (q;q)_n\text(-aq^n;q)_\infty are q-Pochhammer symbols. Gallery References * * *{{dlmf, id=1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Polynomial
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series :y_n(x)=\sum_^n\frac\,\left(\frac\right)^k. Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials :\theta_n(x)=x^n\,y_n(1/x)=\sum_^n\frac\,\frac. The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is :y_3(x)=15x^3+15x^2+6x+1 while the third-degree reverse Bessel polynomial is :\theta_3(x)=x^3+6x^2+15x+15. The reverse Bessel polynomial is used in the design of Bessel electronic filters. Properties Definition in terms of Bessel functions The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name. :y_n(x)=\,x^\theta_n(1/x)\, :y_n(x)=\sqrt\,e^K_(1/x) :\theta_n(x)=\sqrt\,x^e^K_(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Bessel Functions
In mathematics, the incomplete Bessel functions are types of special functions which act as a type of extension from the complete-type of Bessel functions. Definition The incomplete Bessel functions are defined as the same delay differential equations of the complete-type Bessel functions: :J_(z,w)-J_(z,w)=2\dfracJ_v(z,w) :Y_(z,w)-Y_(z,w)=2\dfracY_v(z,w) :I_(z,w)+I_(z,w)=2\dfracI_v(z,w) :K_(z,w)+K_(z,w)=-2\dfracK_v(z,w) :H_^(z,w)-H_^(z,w)=2\dfracH_v^(z,w) :H_^(z,w)-H_^(z,w)=2\dfracH_v^(z,w) And the following suitable extension forms of delay differential equations from that of the complete-type Bessel functions: :J_(z,w)+J_(z,w)=\dfracJ_v(z,w)-\dfrac\dfracJ_v(z,w) :Y_(z,w)+Y_(z,w)=\dfracY_v(z,w)-\dfrac\dfracY_v(z,w) :I_(z,w)-I_(z,w)=\dfracI_v(z,w)-\dfrac\dfracI_v(z,w) :K_(z,w)-K_(z,w)=-\dfracK_v(z,w)+\dfrac\dfracK_v(z,w) :H_^(z,w)+H_^(z,w)=\dfracH_v^(z,w)-\dfrac\dfracH_v^(z,w) :H_^(z,w)+H_^(z,w)=\dfracH_v^(z,w)-\dfrac\dfracH_v^(z,w) Where the new parameter w defines the integral bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel–Maitland Function
In mathematics, the Bessel–Maitland function, or Wright generalized Bessel function, is a generalization of the Bessel function, introduced by . The word "Maitland" in the name of the function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ... seems to be the result of confusing Edward Maitland Wright's middle and last names. It is given by : J^(z) = \sum_ \frac. References * Special functions {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Filter
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat Group delay and phase delay, group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems. The filter's name is a reference to German mathematician Friedrich Bessel (1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949. The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot (signal), overshoot, the Bessel filter has a small amount of overshoot, but still much less than other common frequency-domain filters, such as Butterworth filters. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Beam
A Bessel beam is a wave whose amplitude is described by a Bessel function of the first kind. Electromagnetic, acoustic, gravitational, and matter waves can all be in the form of Bessel beams. A true Bessel beam is non-diffractive. This means that as it propagates, it does not diffract and spread out; this is in contrast to the usual behavior of light (or sound), which spreads out after being focused down to a small spot. Bessel beams are also ''self-healing'', meaning that the beam can be partially obstructed at one point, but will re-form at a point further down the beam axis. As with a plane wave, a true Bessel beam cannot be created, as it is unbounded and would require an infinite amount of energy. Reasonably good approximations can be made, however, and these are important in many optical applications because they exhibit little or no diffraction over a limited distance. Approximations to Bessel beams are made in practice either by focusing a Gaussian beam with an axicon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Process
In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. Formal definition The Bessel process of order ''n'' is the real-valued process ''X'' given (when ''n'' ≥ 2) by :X_t = \, W_t \, , where , , ·, , denotes the Euclidean norm in R''n'' and ''W'' is an ''n''-dimensional Wiener process (Brownian motion). For any ''n'', the ''n''-dimensional Bessel process is the solution to the stochastic differential equation (SDE) :dX_t = dW_t + \frac\frac where W is a 1-dimensional Wiener process (Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...). Note that this SDE makes sense for any real parameter n (although the drift term is singular at zero). Notation A notation for the Bessel process of dimension started at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
