What you want is a variable frequency drive, or VFD. This is a commodity part, easily available from a wide range of manufacturers and distributors. At the power levels you're describing, many are specifically designed to take single-phase AC in and put out three-phase AC out to a motor.
A quick Google search found these:
http://www.vfds.com/3-hp-vfd/3hp-230v-mitsubishi-vfd-frd720s100na
http://www.automationdirect.com/adc/Overview/Catalog/Drives
http://www.amazon.com/Ten-high-3000w-Variable-Frequency-Inverter/dp/B00BKWKR50/ref=sr_1_1?ie=UTF8&qid=1445866136&sr=8-1&keywords=vfd
Now, if you actually want to design such a thing, that's much more complicated! You'll need to know how to design a rectifier, both electrically and thermally. You'll need to know the same about an inverter. You'll need to size the capacitors properly. You'll need a precharge. And that's before you even get into how you control a motor by feeding it voltage and frequency. This is very much not a trivial endeavor; there are several billion-dollar companies who do this for a living!
Not to say you can't or shouldn't do it. Just know that it's an investment of months or years into making something that you can buy for $300. If you succeed, you'll be much smarter! Just make sure you understand that that's your goal when you start, and not necessarily to be cost-effective.