The　paper　develops　and　examines　the　complete　solutions　for　the　elastic　field　induced　by　the　point　load　vector　in　a　general　functionally　graded　material　(FGM)　model　with　transverse　isotropy.　The　FGMs　are　approximated　with　n-layered　materials.　Each　of　the　n-layered　materials　is　homogeneous　and　transversely　isotropic.　The　complete　solutions　of　the　displacement　and　stress　fields　are　explicitly　expressed　in　the　forms　of　fifteen　classical　Hankel　transform　integrals　with　ten　kernel　functions.　The　ten　kernel　functions　are　explicitly　expressed　in　the　forms　of　backward　transfer　matrices　and　have　clear　mathematical　properties.　The　singular　terms　of　the　complete　solutions　are　analytically　isolated　and　expressed　in　exact　closed　forms　in　terms　of　elementary　harmonic　functions.　Numerical　results　show　that　the　computation　of　the　complete　solutions　can　be　achieved　with　high　accuracy　and　efficiency.