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 formula_1 is the representation of that function as a linear combination of many orthogonal versions of the same Bessel function of the first kind "J", where the argument to each version "n" is differently scaled, according to formula_2 where "u" is a  root, numbered "n" associated with the Bessel function "J" and "c" are the assigned coefficients: formula_3 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. As said, differently scaled Bessel Functions are orthogonal with respect to the inner product formula_4 according to formula_5 (where: formula_6 is the  Kronecker delta). The coefficients can be obtained from projecting the function onto the respective Bessel functions: formula_7 where the plus or minus sign is equally valid. Application. 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, discrimination of odorants in a turbulent ambient, postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, EEG signal segmentation, speech enhancement, and speaker identification. The Fourier–Bessel series expansion has also been used to  reduce cross terms in the Wigner–Ville distribution. Dini series. A second Fourier–Bessel series, also known as "Dini series", is associated with the Robin boundary condition formula_8 where formula_9 is an arbitrary constant. The Dini series can be defined by formula_10 where formula_11 is the "n"-th zero of formula_12. The coefficients formula_13 are given by formula_14