The discovery of the first stone arrowheads in Africa tends to indicate that the bow and arrow were invented there, maybe as early as 50,000 BC.
It was probably developed in conjunction with the invention of the spear thrower.
A short bow would be a better hunting weapon when used to stalk animals in wooded areas, rather than carry around long spears.
The shape of the earliest bows can only be guessed at, as broken or worn out bows would probably end up on the cooking fire, reshaped into different tools or just thrown away.