The　paper　develops　the　Green＇s　function　solution　in　matrix　form　for　functionally　graded　and　transversely　isotropic　material　(FGTIM)　due　to　the　loading　of　a　circular　ring　force　vector.　The　force　vector　is　concentrated　along　a　horizontal　circular　ring　and　acts　at　any　location　in　the　interior　of　the　FGTIM　material　occupying　either　a　fullspace　or　a　halfspace.　Without　loss　of　generality,　the　variations　of　the　FGTIM＇s　five　elastic　parameters　are　expressed　as　five　arbitrary　step　functions　with　the　depth　as　the　independent　variable.　The　two-dimensional　Fourier　integral　transforms　in　the　cylindrical　coordinate　system　are　used　for　the　mathematical　formulation　and　derivation　in　matrix　form.　The　Green＇s　function　solution　of　the　displacement　and　stress　fields　is　explicitly　expressed　in　the　matrix　form　in　terms　of　classical　improper　Hankel　transform　integrals　of　the　orders　of　0　to　3.　The　kernel　functions　of　the　Hankel　transform　integrals　are　explicitly　expressed　in　the　forms　of　backward　transfer　matrix.　Their　mathematical　properties　are　analyzed　analytically　and　numerically.　The　singular　terms　associated　with　the　improper　Hankel　transform　integrals　are　analytically　isolated　and　expressed　in　the　exact　closed-form　in　terms　of　the　complete　elliptic　integrals　of　the　first,　second　and　third　kind.　Such　closed-form　singular　terms　can　be　expressed　as　the　matrix　Green＇s　function　solution　for　the　corresponding　bi-material　due　to　the　same　circular　ring　force　vector.　Numerical　results　show　that　the　computation　of　the　matrix　Green＇s　function　solution　can　be　ach-ieved　with　high　accuracy　and　efficiency　and　the　heterogeneity　and　anisotropy　can　have　significant　effects　on　the　elastic　fields　in　the　FGTIM　induced　by　the　circular　ring　force　vector.