What is the use of private static member functions?

Question

I was looking at the request parser from the boost::asio example and I was wondering why the private member functions like is_char() are static? :

Answer 1

So is there a reason why these functions are static?

Non-static member functions have a hidden additional parameter called this. Passing this doesn't come for free, so making a private function static can be seen as a means of optimization.
But it can also be seen as a means of expressing your requirements/design in your code: If that function doesn't need to refer to any member data of the class, why should it be a non-static member function?

However, changing the type of any member function, public or private, static or not, will require all clients to recompile. If this needs to be done for a private function which those clients can never use, that's a waste of resources. Therefore, I usually move as many functions as possible from the class' private parts into an unnamed namespace in the implementation file.