Editing Bessel function (section)

=== Waves and elasticity problems ===
The first appearance of a Bessel function appears in the work of [[Daniel Bernoulli]] in 1732, while working on the analysis of a [[String vibration|vibrating string]], a problem that was tackled before by his father [[Johann Bernoulli]].<ref name=":0" /> Daniel considered a flexible chain suspended from a fixed point above and free at its lower end.<ref name=":0" /> The solution of the differential equation led to the introduction of a function that is now considered  <math>J_0(x)</math>. Bernoulli also developed a method to find the zeros of the function.<ref name=":0" />

[[Leonhard Euler]] in 1736, found a link between other functions (now known as [[Laguerre polynomials]]) and Bernoulli's solution. Euler also introduced a non-uniform chain that lead to the introduction of functions now related to modified Bessel functions <math>I_n(x)</math>.<ref name=":0" /> 

In the middle of the eighteen century, [[Jean le Rond d'Alembert]] had found a [[D'Alembert's formula|formula]] to solve the [[wave equation]]. By 1771 there was dispute between Bernoulli, Euler, d'Alembert and [[Joseph-Louis Lagrange]] on the nature of the solutions vibrating strings.<ref name=":0" />

Euler worked in 1778 on [[buckling]], introducing the concept of [[Euler's critical load]]. To solve the problem he introduced the series for <math>J_{\pm 1/3}(x)</math>.<ref name=":0" /> Euler also worked out the solutions of vibrating 2D membranes in cylindrical coordinates in 1780. In order to solve his differential equation he introduced a power series associated to <math>J_n(x)</math>, for integer ''n''.<ref name=":0" />

During the end of the 19th century Lagrange, [[Pierre-Simon Laplace]] and [[Marc-Antoine Parseval]] also found equivalents to the Bessel functions.<ref name=":0" /> Parseval for example found an integral representation of <math>J_0(x)</math> using cosine.<ref name=":0" />

At the beginning of the 1800s, [[Joseph Fourier]] used <math>J_0(x)</math> to solve the [[heat equation]] in a problem with cylindrical symmetry.<ref name=":0" /> Fourier won a prize of the [[French Academy of Sciences]] for this work in 1811.<ref name=":0" /> But most of the details of his work, including the use of a [[Fourier series]], remained unpublished until 1822.<ref name=":0" /> Poisson in rivalry with Fourier, extended Fourier's work in 1823, introducing new properties of Bessel functions including Bessel functions of half-integer order (now known as spherical Bessel functions).<ref name=":0" />