It is shameful that Donald Trump has seized on the question of President Obama's birth certificate as a means of pandering to the right wing of the Republican party. Like most things he does, it's a publicity stunt, but it is one that is part of a plan that aims at a thought almost too terrible to consider, the first stage of which is to get enough support among Republicans that he can win primaries and delegates.
I believe that Obama was born in Hawaii, and that Hawaii was part of U.S. territory at the time (one would have to go back to the legality of the resolution in 1895 that made it so, and it was truly an abuse of power, but not illegal under U.S. law). His father seems to have been a Kenyan who had some Muslims among his forebears, but it is ridiculous to think that religion is specified on a birth certificate, so what is it they are looking for?
The whole "investigation", to dignify it overly, is wrong-headed. It actually doesn't matter, for Obama's legal qualification to be President, where he was born. He was born of a native-born (white) American mother, so he's a native citizen and eligible for the Presidency, no matter where the birth occurred. In order to question his status, one would have to establish he was not borne by his mother.
End of story.