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Bessel function - mklabgr - 08-10-2026
Bessel function
Summary
Bessel functions are a family of special functions that arise as solutions to Bessel’s differential equation, $x^2y''+xy'+(x^2-\alpha^2)y=0$, and are especially important for problems with circular, cylindrical, or spherical symmetry.
They were systematically studied by the German mathematician and astronomer Friedrich Bessel and are widely used in physics and engineering, including wave propagation, heat conduction, electromagnetic waves, vibrations of circular membranes, acoustics, quantum mechanics, fluid dynamics, and signal processing.
The main types include Bessel functions of the first kind $J_\alpha(x)$, second kind $Y_\alpha(x)$, Hankel functions, modified Bessel functions, and spherical Bessel functions. The functions of the first kind generally exhibit oscillatory behavior similar to sine and cosine, with their amplitude decreasing approximately as $x^{-1/2}$ for large $x$, while their mathematical properties include recurrence relations, integral representations, series expansions, and important patterns in their zeros.
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