Bessel

Bessel most commonly refers to Friedrich Wilhelm Bessel (1784–1846), a German astronomer and mathematician known for precise stellar parallax measurements and contributions to celestial mechanics. In mathematics and physics, the name is attached to Bessel functions, a family of solutions to differential equations that model problems with cylindrical symmetry. The functions J_n and Y_n (and their modified counterparts I_n and K_n) are named after him and appear widely in wave propagation, heat conduction, and static potentials in cylindrical coordinates.

Bessel's differential equation is x^2 y'' + x y' + (x^2 - n^2) y = 0. Its regular solutions J_n(x), called Bessel functions of the first kind, are finite at the origin for integer n. The linearly independent second solution Y_n(x), the Neumann function, is singular at the origin. For problems with exponential growth, the modified Bessel equations x^2 y'' + x y' - (x^2 + n^2) y = 0 yield I_n and K_n.

Bessel functions are characterized by series expansions, integral representations, and recurrence relations. They arise from separating variables in the Helmholtz and Laplace equations in cylindrical coordinates, and they describe modes of a circular drum, cylindrical waveguides, and radial heat conduction. They also appear in signal processing as kernels in the Fourier-Bessel, or Hankel, transforms.

In applied contexts, the zeros of J_n determine eigenfrequencies in bounded cylindrical systems, and orthogonality relations of J_n allow expansions of radial functions into Bessel series. As such, Bessel functions are standard tools in physics, engineering, and applied mathematics.

The name Bessel is used in other contexts as well, including the crater Bessel on the Moon, named in his honor.

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