If the elven wizard had proficiency in a bow could he use that as his arcane focus?
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Note: This is my second answer on this question. It uses much of the same information as my first answer but comes to a different conclusion. I wanted to make sure that votes for my other answer did not erroneously count against this answer. Both answers feel accurate to me, based on slightly different points of view.
It's not called out, but there's precedent: go for it
D&D Basic Rules V0.2, page 30
D&D Basic Rules V0.2, page 47
The Adventuring Gear table on page 48 lists 5 possibilities for an Arcane Focus (which mirrors the list on page 47): Crystal, Orb, Rod, Staff, Wand. None of these are bows. None of these even appear to be weapons: initially, I'd rule, "No, it's not possible per RAW."
However, while researching Can the arcane/druidic focus staff double as quarterstaff?, I asked Mike Mearls: "Can the arcane/druidic focus staff double as a quarterstaff?" And received the following response: "yes"
This means that there is precedent for allowing an Arcane Focus as a weapon. Notice, however, that the Arcane Focus Staff is 25 times more expensive than a plain Quarterstaff: make sure your Arcane Focus Bow is significantly more expensive than a regular bow.
Not Per the RAW.
D&D Basic Rules V0.1, page 30
D&D Basic Rules V0.1, page 48
The Adventuring Gear table on page 47 lists 5 possibilities for an Arcane Focus (which mirrors the list on page 48): Crystal, Orb, Rod, Staff, Wand. None of these are bows. None of these are even weapons.
Wait, the staff is a weapon, so there's precedent!
I don't care about the RAW, what are my options?
I think, going along with what is in the rules for 5e, that if, for example, your bow had a crystal embedded in it's wood, you could use the crystal as your focus, and it would technically be the bow, but still follow rules. Just a suggestion.
An Arcane Foci is an quasi magical object that has been specially tuned for spell casting. Such an item would be specifically and painstakingly attuned to the spell caster. If it is made for the character It would make sense then that any damage, or abnormality to the foci would render it inert. Much like a fighter must maintain his weapon through sharpening and oiling, a mage would have to repair damage (such as re-carving runes damaged in a battle) or purge impure or alien energy before the foci becomes useful again. While a Foci could be used for combat, it would likely be a last ditch effort as repair would take time and money.