This　paper　investigates　the　contact　problem　of　a　layered　elastic　halfspace　with　transverse　isotropy　under　the　axisymmetric　indentation　of　a　circular　rigid　plate.　Fourier　integral　transforms　and　a　backward　transfer　matrix　method　are　used　to　obtain　the　analytical　solution　of　the　contact　problem.　The　interaction　between　the　rigid　plate　and　the　layered　halfspace　can　be　expressed　with　the　standard　Fredholm　integral　equations　of　the　second　kind.　The　induced　elastic　field　in　the　layered　halfspace　can　be　expressed　as　the　semi-infinite　integrals　of　four　known　kernel　functions.　The　convergence　and　singularity　of　the　semi-infinite　integrals　near　or　at　the　surface　of　the　layered　halfspace　are　resolved　using　an　isolating　technique.　The　efficient　numerical　algorithms　are　used　and　developed　for　accurately　calculating　the　Fredholm　integral　equations　and　the　semi-infinite　integrals.　Numerical　results　show　the　correctness　of　the　proposed　method　and　the　effect　of　layering　non-homogeneity　on　the　elastic　fields　in　layered　transversely　isotropic　halfspace　induced　by　the　axisymmetric　indentation　of　a　circular　rigid　plate.