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This secton is devoted to one of the most important differential equations---Bessel equation. Its solutions were named for Friedrich Wilhelm Bessel
(1784--1846); however, Daniel Bernoulli is generally credited with being the first to introduce the concept of Bessels functions in 1732. He used the function of zero order as a solution to the problem of an oscillating
chain suspended at one end. In 1764 Leonhard Euler employed Bessel functions of both zero and integral orders in an analysis of vibrations of a stretched membrane, an investigation which was further developed by Lord Rayleigh in 1878, where he demonstrated that Bessels
functions are particular cases of Laplaces functions.
Bessel, while receiving named credit for these functions, did not incorporate them into his work as an astronomer until 1817. The Bessel function was the result of Bessels study of a
problem of Kepler for determining the motion of three bodies moving under mutual gravita-
tion. In 1824, he incorporated Bessel functions in a study of planetary perturbations where
the Bessel functions appear as coefficients in a series expansion of the indirect perturbation
of a planet, that is the motion of the Sun caused by the perturbing body. It was likely
Lagrange’s work on elliptical orbits that first suggested to Bessel to work on the Bessel functions.
The notation J_{z,n} was first used by Hansen (1843) and subsequently by Schlomilch (1857) and later modified to J_{n}(2z) by Watson (1922). Subsequent studies of Bessel functions included the works of Mathews in 1895, “A treatise
on Bessel functions and their applications to physics” written in collaboration with Andrew Gray. It was the first major treatise on Bessel functions in English and covered topics such as applications of Bessel functions to electricity, hydrodynamics and diffraction. In 1922, Watson first published his comprehensive examination of Bessel functions “A Treatise on the Theory of Bessel Functions”.
where ν is a real constant, called the order of the Bessel equation. Upon substitution \( y = u\, x^{-1/2} , \) it is reduced to the equation without first derivative:
where j is the unit vector in positive vertical direction on the complex plane ℂ.
Because of the linear independence of the Bessel function of the first and second
kind, the Hankel functions provide an alternative pair of solutions to the Bessel
differential equation.
================================ to be modified ==============
where Γ is the Gamma function of Euler. The above series (which converges for all real or complex x) defines the Bessel function of the first kind. If \( \nu \) is not an integer, then the function \( J_{-\nu} (x) \) is linearly independent of \( J_{\nu} (x) . \) Therefore these two functions constitute the fundamental set of solutions to Bessel's equation. When \( \nu =n \) is an integer,
\[
J_{-n} (x) = (-1)^n \,J_{n} (x) ,
\]
and we need to find another linearly independent solution to Bessel's equation. This can be done using reduction of order, which leads to another linearly independent solution:
Upon multiplication by a suitable constant, this expression defines either \( Y_n (x) = \pi N_n (x), \) the Weber function, or \( N_n (x) , \) the Neumann function. Heinrich Martin Weber (1842--1913) and Carl (also Karl) Gottfried Neumann (1832--1925) were German mathematicians. The Neumann function can be defined by a single expression:
where \( \gamma = \Gamma '(1) \approx 0.5772156 \ldots \) is the Euler constant. Bessel functions can be expressed through the single Bessel function of order zero:
where T_{n} is the Chebychev polynomial of the first kind and j is the unit vector in positive vertical direction on the complex plane. The parametric Bessel differential equation
Grigorieva, E., Methods of Solving Sequence and Series Problems, Birkhäuser; 1st ed. 2016.
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