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 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.

Because Bessel functions are orthogonal with respect to a weight function x on the interval [0,b], they can be expanded in a Fourier–Bessel series defined by

$f(x) \sim \sum_{n=0}^\infty c_n J_\alpha(\lambda_n x/b)$,

where λ_n is the nth zero of J_α(x). This series is associated with the boundary condition f(b) = 0.

From the orthogonality relationship

$\int_0^1 J_\alpha(x \lambda_m)\,J_\alpha(x \lambda_n)\,x\,dx = \frac{\delta_{mn}}{2} [J_{\alpha+1}(\lambda_n)]^2$,

the coefficients are given by

$c_n =\frac{\int_{0}^b J_\alpha(\lambda_n x/b)\,f(x) \,x\,dx }{\int_{0}^b x J_\alpha^2 (\lambda_n x/b) dx} =\frac{\langle f, J_\alpha(\lambda_n x/b) \rangle}{\|J_\alpha(\lambda_n x/b)\|^2}.$

The lower integral may be evaluated, yielding

$c_n =\frac{\int_{0}^b J_\alpha(\lambda_n x/b)\,f(x) \,x\,dx }{b^2 J_{\alpha\pm 1}^2 (\lambda_n)/2}$,

where the plus or minus sign is equally valid.

Dini series

A second Fourier–Bessel series, also known as Dini series, is associated with the Robin boundary condition

bf'(b) + cf(b) = 0, where c is an arbitrary constant.

The Dini series can be defined by

$f(x) \sim \sum_{n=0}^\infty b_n J_\alpha(\gamma_n x/b)$,

where γ_n is the nth zero of xJ'_α(x) + cJ_α(x).

The coefficients b_n are given by

$b_n = \frac{2 \gamma_n^2}{ b^2(c^2+\gamma_n^2-\alpha^2)J_\alpha^2(\gamma_n)} \int_{0}^b J_\alpha(\gamma_n x/b)\,f(x) \,x\,dx$.

References

• Smythe, William R. (1968). Static and Dynamic Electricity (3rd ed.). New York: McGraw-Hill. 

• Magnus, Wilhelm; Oberhettinger, Fritz; Soni, Raj Pal (1966). Formulas and Theorems for Special Functions of Mathematical Physics. Berlin: Springer. 

External links

• Weisstein, Eric. W. "Fourier-Bessel Series". From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Fourier-BesselSeries.html. 
• Fourier–Bessel series applied to Acoustic Field analysis on Trinnov Audio's research page

See also

• Orthogonality
• Generalized Fourier series
• Hankel transform
• Neumann polynomial