In this note, we get the Fourier expansion for the non-holomorphic Eisenstein series by slight modification of Maass’ original method, which enables us to prove as a bonus, two integral representations for the modified Bessel functionof the third kind.This kind revealsahidden inner structureofthe non-holomorphic Eisenstein series and the Bessel diffenrential equation. We also explain a work of Motohashi on the Kronecker limit formula for the Epstein zeta-function from our point of view. © 2010 Taylor & Francis.