Integral Representation for Bessel’s Functions of the First Kind and Neumann Series

  • Enrico De Micheli


A Fourier-type integral representation for Bessel’s functions of the first kind and complex order is obtained by using the Gegenbauer extension of Poisson’s integral representation for the Bessel function along with a suitable trigonometric integral representation of Gegenbauer’s polynomials. By using this representation, expansions in series of Bessel’s functions of various functions which are related to the incomplete gamma function can be obtained in a unified way. Neumann series are then considered and a new closed-form integral representation for this class of series is given. The density function of this representation is the generating function of the sequence of coefficients of the Neumann series on the unit circle. Examples of new closed-form integral representations of special functions are thus presented.


Bessel function of the first kind Neumann series integral representation special function 

Mathematics Subject Classification

Primary 33C10 40C10 Secondary 33B20 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IBF - Consiglio Nazionale delle RicercheGenovaItaly

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