business use of home

Yes

"Can he take a business use of home for the expenses incurred during the first half of the year"

"Second half he was both self employed"

He can take expenses for all year, for those expenses that correlate to the self-employed income. So, not everything, but anything severable or which can be allocated.

Of course he can claim deduction for first half of year, if he otherwise qualifies.  For the second half, can he have exclusive use if he is also using it as an employee?  Well, that is a trade or business, just not one for which office in home deductions are allowed. (Or are they?  Aren't there special rules for musicians and other artists?)  

So maybe if 80% of the time in the second half, it is used for self-employment, then claim 80% of the expenses, because the other 20% is connected to a trade or business where the deduction is not allowed?  Or should you allocate based on income earned from each activity?  Does it really matter, since IRS isn't going to be doing any audits for a while anyway?