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Fourier–Bessel series
Infinite series of Bessel functions From Wikipedia, the free encyclopedia
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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.
![]() | This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2014) |
Fourier–Bessel series are used in the solution to partial differential equations, particularly in cylindrical coordinate systems.
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Definition
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The Fourier–Bessel series of a function f(x) with a domain of [0, b] satisfying f(b) = 0

is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind Jα, where the argument to each version n is differently scaled, according to [1][2] where uα,n is a root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients:[3]
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Interpretation
The Fourier–Bessel series may be thought of as a Fourier expansion in the ρ coordinate of cylindrical coordinates. Just as the Fourier series is defined for a finite interval and has a counterpart, the continuous Fourier transform over an infinite interval, so the Fourier–Bessel series has a counterpart over an infinite interval, namely the Hankel transform.
Calculating the coefficients
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As said, differently scaled Bessel Functions are orthogonal with respect to the inner product

according to
(where: is the Kronecker delta). The coefficients can be obtained from projecting the function f(x) onto the respective Bessel functions:
where the plus or minus sign is equally valid.
For the inverse transform, one makes use of the following representation of the Dirac delta function[4]
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Applications
The Fourier–Bessel series expansion employs aperiodic and decaying Bessel functions as the basis. The Fourier–Bessel series expansion has been successfully applied in diversified areas such as Gear fault diagnosis,[5] discrimination of odorants in a turbulent ambient,[6] postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, speech enhancement,[7] and speaker identification.[8] The Fourier–Bessel series expansion has also been used to reduce cross terms in the Wigner–Ville distribution.
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Dini series
A second Fourier–Bessel series, also known as Dini series, is associated with the Robin boundary condition where is an arbitrary constant. The Dini series can be defined by
where is the n-th zero of .
The coefficients are given by
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See also
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