Who invented the first semi auto gun?

I don't know for sure, but here are some suggestions and starting points to look at with your own research. All of the people I'm about to name had popular mass-produced semi-auto pistols. As you know there is often a big difference between when a certain kind of thing is "invented" and when it becomes popular and available for the public to buy.

Hugo Borchardt had a semi-automatic pistol in the early 1890s.

This gun was the inspiration for the famous Luger pistol, invented by Georg Luger in the early 1900s.

Paul and Wilhelm Mauser produced a semi-automatic (and fully automatic too) military pistol called the C96 Mauser or "broom-handle" Mauser back in the middle 1890s.

John Moses Browning designed a number of semi-auto handguns, both small and large, in the late 1890s through the WWI years. The famous Army model 1911 pistol is what he was most famous for, but an earlier version of this kind of gun, in a smaller caliber, came on the market in 1903, and it was very successful. Browning designed a lot of guns for the Colt company.