Do you have to serve the ball by throwing it overhead or can you actually serve underhand or side arm?
I believe you can serve it underhand, but the reason no one does this is that you would have to hit it slowly, so that the ball after going over the net would have time to drop down into the zone. By hitting it overhand, and reaching up as high as you can, the ball from that point has basically a straight-line view of the zone, so you can hit it as hard as you please and still be a good serve.
Yes, you are allowed to serve underhand or sidearm.
Rules
Michael Chang, who retired a couple days ago, was so weary and cramped during the 1989 French Open quarter-finals, he served side armed. Won the point, too.
Won the tournament, actually.
FTR, this isn’t quite true. Unless you’re about 6’7" or taller, the straight-line from the top of your racket and just over the net ends up a bit long for a good serve. You’ll need to either take some speed off (eww!) or add some topspin, serving a ‘curve ball’, as it were … But this is still the right basic answer. The lower your impact point on the serve, the more speed you’ll need to lose/the more topspin you’ll need to add, and the easier your serve will be to return.
It’s generally considered bad form (and somewhat rude to the opposition) unless you are seriously exhausted, but it is legal.
Hingis did it while throwing a tantrum about losing a Slam (the French?) to Graf. She was soundly booed for it.