Bessel generating functions
Peter Andreas Hansen (1795--1874), a German astronomer, was the first who discovered in 1843 the generating function for the Bessel functions of the first kind:
\[
e^{x \left( z - 1/z \right) /2} = \sum_{n=-\infty}^{\infty} z^n J_n (x)
\]
Hansen became director of the Seeberg Observatory, near Gotha, in 1825, and in 1857 a new observatory was built for him. He worked on theoretical geodesy, optics, and probability theory. His most important books on the theory of the motion of the Moon are the Fundamenta nova investigationis orbitae verae quam Luna perlustrat (1838; “New Foundations of the Investigation of the True Orbit That the Moon Traverses”) and the Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Störungen (1862–64; “Explanation of the Theoretical Calculation of Perturbations Used in Lunar Tables”). The systematic character of Hansen’s methods carried celestial mechanics to a new level of power and precision.
- Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4.
- Dutka, J., On the early history of Bessel functions, Archive for History of Exact Sciences, volume 49, pages 105–134 (1995). https://doi.org/10.1007/BF00376544
- Watson, G.N., A Treatise on the Theory of Bessel Functions,
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