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Concept
Fourier–Bessel series
Summary
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 f(x) with a domain of satisfying f(b) = 0
f: [0,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 Jα, where the argument to each version n is differently scaled, according to
(J_\alpha )n (x) := J\alpha \left( \frac{u_{\alpha,n}}b x \right)
where uα,n is a root, numbered n associated with the Bessel function Jα and cn are the assigned coefficients:
f(x) \sim \sum_{n=1}^\infty c_n J_\alpha \left( \frac{u_{\alpha,n}}b x \right).
Interpret
Official source
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