I've raised cattle, goats, horses, sheep and chickens, and I've had trouble with dogs from time to time. You do need to have some fences, and the tighter the better. That is just good sense. It will keep you animals in (mostly) and keep dogs and other predators out (mostly).
If the distances are fairly short, I would get a CO2 powered BB gun (
http://www.pyramydair.com/s/m/Umarex_Steel_Storm_CO2_Gun/2043), and hopefully you could discourage the dog with little or no permanent damage. If not BB, then airsoft (
http://www.pyramydair.com/s/m/Tactical_Force_TF11_CO2_Airsoft_Submachine_Gun/1421).
If you can't reach an accommodation with your neighbor, you need to decide if it is worth the fight to protect your animals. Maybe it is just easier and better to raise a garden and some fruit trees. If you don't want to give up your animals, check the laws of your state to see what you can legally do if a dog is (1) on your property, (2) watching your stock, or (3) harassing your stock. Laws very from state to state. In Colorado, and Utah, the states I'm most familiar with, you can shoot a dog if it is watching or harassing your stock.
Maybe this is too abhorrent for you to contemplate. Then you need to reevaluate how badly you want to keep stock. If you are keeping them, they deserve your protection. I'm not wild about shooting dogs, but I've seen some bad damage that they can cause that kind of hardened my heart. I came home one night from a swing shift and found a little yearling ewe with all the meat chewed off of her hind leg, clear up to the hip. Just bones left, and she was still alive. We had raised her from a baby with a bottle, and it wasn't a great experience putting her down.
If you take this path, I would also advise that you keep pretty quiet about it. My motto is SSS - shoot, shovel, shut-up. We live in an upside down world as far as I can determine. People can't or won't take care of their pets, but often are totally oblivious to the damage and trouble they cause. To them, their pets are their children..... I don't even know how to address this.
Good luck.
Rune