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Bessel function | A Wisdom Archive on Bessel function | | Bessel function A selection of articles related to Bessel function | |
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More material related to Bessel Function can be found here:
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| | | Bessel function, Bessel function - Applications, Bessel function - Asymptotic forms, Bessel function - Definitions, Bessel function - Properties, Bessel function - Bessel functions of the first kind, Bessel function - Bessel functions of the second kind, Bessel function - Hankel functions, Bessel function - Modified Bessel functions, Bessel function - Riccati-Bessel functions, Bessel function - Spherical Bessel functions | | | » Page 1 « Page 2 | |
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| ARTICLES RELATED TO Bessel function | |
| | |  Bessel function: Encyclopedia II - Bessel function - Definitions
Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below.
Bessel function - Bessel functions of the first kind.
Bessel functions of the first kind, denoted with Jα(x), are solutions of Bessel's differential equation which are finite at x = 0 for α an integer or α non-negative. The s ...
See also:Bessel function, Bessel function - Applications, Bessel function - Definitions, Bessel function - Bessel functions of the first kind, Bessel function - Bessel functions of the second kind, Bessel function - Hankel functions, Bessel function - Modified Bessel functions, Bessel function - Spherical Bessel functions, Bessel function - Riccati-Bessel functions, Bessel function - Asymptotic forms, Bessel function - Properties Read more here: » Bessel function: Encyclopedia II - Bessel function - Definitions |
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| | | | Bessel function: Encyclopedia II - Bc programming language - POSIX bc The POSIX standardised bc language is traditionally written as a program in the dc programming language to provide a higher level of access to the features of the dc language without the complexities of dc's terse syntax.
In this form, the bc language contains single letter variable, array and function names and most standard arithmetic operators as well as the familiar control flow constructs, (if(cond)..., while(cond)... and for(init;cond;inc)...) from C. Unlike C, an if clause ...
See also:Bc programming language, Bc programming language - POSIX bc, Bc programming language - Mathematical operators, Bc programming language - Built-in functions, Bc programming language - Standard library functions, Bc programming language - GNU bc, Bc programming language - Extra operators, Bc programming language - Functions, Bc programming language - Example code, Bc programming language - A 'Power' function in POSIX bc, Bc programming language - An equivalent 'Power' function in GNU bc Read more here: » Bc programming language: Encyclopedia II - Bc programming language - POSIX bc |
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| | | | Bessel function: Encyclopedia II - Dirac delta function - Overview Dirac functions can be of any size in which case their 'strength' A is defined by duration multiplied by amplitude. The graph of the delta function can be usually thought of as following the whole x-axis and the positive y-axis. (This informal picture can sometimes be misleading, for example in the limiting case of the sinc function.)
Despite its name, the delta function is not a function as defined in the strictest mathematical sense. One reason for this is because the functions f(x) = δ(x) and ...
See also:Dirac delta function, Dirac delta function - Overview, Dirac delta function - Formal introduction, Dirac delta function - Delta function of more complicated arguments, Dirac delta function - Fourier transform, Dirac delta function - The Dirac delta function as a probability density function, Dirac delta function - Derivatives of the delta function, Dirac delta function - Equivalent definition, Dirac delta function - Representations of the delta function Read more here: » Dirac delta function: Encyclopedia II - Dirac delta function - Overview |
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| | | | Bessel function: Encyclopedia II - Aliasing - Mathematical explanation of aliasing The preceding explanation and the Nyquist criterion are somewhat idealised, because they assume instantaneous sampling and other slightly unrealistic hypotheses, although useful approximations to these things do exist. The following is a more detailed explanation of the phenomenon in terms of function approximation theory.
Aliasing - Continuous signals.
For the purposes of this analysis, we define (continuous) signal as a real or complex valued function whose domain is the interval [0,1]. To ...
See also:Aliasing, Aliasing - Overview, Aliasing - Aliasing in periodic phenomena, Aliasing - Sampling a sinusoidal signal, Aliasing - The Nyquist criterion, Aliasing - Origin of the term, Aliasing - An audio example, Aliasing - Mathematical explanation of aliasing, Aliasing - Continuous signals, Aliasing - Point sampling, Aliasing - A better sampling method filtering, Aliasing - Reconstruction, Aliasing - Aliasing, Aliasing - Optimal filtering, Aliasing - Caveats, Aliasing - An example in astronomy, Aliasing - External link Read more here: » Aliasing: Encyclopedia II - Aliasing - Mathematical explanation of aliasing |
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| | | | Bessel function: Encyclopedia II - Dirac delta function - Fourier transform The continuous Fourier transform of the Dirac delta is the constant function . The inverse transform of this constant function will be the Dirac delta again, yielding the orthogonality property for the Fourier kernel:
From the convolution theorem for the Fourier transform, the convolution of δ with any distribution S yields S.
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See also:Dirac delta function, Dirac delta function - Overview, Dirac delta function - Formal introduction, Dirac delta function - Delta function of more complicated arguments, Dirac delta function - Fourier transform, Dirac delta function - The Dirac delta function as a probability density function, Dirac delta function - Derivatives of the delta function, Dirac delta function - Equivalent definition, Dirac delta function - Representations of the delta function Read more here: » Dirac delta function: Encyclopedia II - Dirac delta function - Fourier transform |
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| | | | Bessel function: Encyclopedia II - Dirac delta function - Delta function of more complicated arguments A helpful identity is the scaling property:
and so
This concept may be generalized to:
where xi are the roots of g(x). In the integral form it is equivalent to
In an n-dimensional space with position vector , this is generalized to:
where the integral on the right is over , the n-1 dimensional surface defined by .
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See also:Dirac delta function, Dirac delta function - Overview, Dirac delta function - Formal introduction, Dirac delta function - Delta function of more complicated arguments, Dirac delta function - Fourier transform, Dirac delta function - The Dirac delta function as a probability density function, Dirac delta function - Derivatives of the delta function, Dirac delta function - Equivalent definition, Dirac delta function - Representations of the delta function Read more here: » Dirac delta function: Encyclopedia II - Dirac delta function - Delta function of more complicated arguments |
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| | | | Bessel function: Encyclopedia II - Dirac delta function - Representations of the delta function The delta function can be viewed as the limit of a sequence of functions
where δa(x) is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting sequence will be used. On the other hand the term limit needs to be made precise, as this equality holds only for some meanings of limit. The term approx ...
See also:Dirac delta function, Dirac delta function - Overview, Dirac delta function - Formal introduction, Dirac delta function - Delta function of more complicated arguments, Dirac delta function - Fourier transform, Dirac delta function - The Dirac delta function as a probability density function, Dirac delta function - Derivatives of the delta function, Dirac delta function - Equivalent definition, Dirac delta function - Representations of the delta function Read more here: » Dirac delta function: Encyclopedia II - Dirac delta function - Representations of the delta function |
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