Abstract: In cylindrical coordinates，the family of intrinsic Bessel function constitutes a complete orthogonal function series，which can be used as the bases of generalized Fourier expansion. In this paper，starting from the generalized Fourier expansion of a function defined on a finite interval and using the approximate formula of Bessel function and its zero point formula，we discuss the Fourier-Bessel integral expansion of a function defined in semiinfinite space，and get the approximate expression of module square of Bessel function. In the asymptotical situation， discontinuous parameter becomes continuous one，we obtain the Fourier-Bessel integral and coefficient formula of the function.

Key words: Bessel function, Fourier-Bessel integral