Consider a ket space spanned by the eigen kets {|a_i}of a Hermitian operator A, i.e., A|ai= a_i | a_i. There is no degeneracy.

Now take A = a + b where a is a real number, b is a real vector (a, b are not operators), and is a vector operator whose Cartesian components are Pauli matrices x, y, and z.

Need to Prove:

f(a + b ) = 1/2 { f(a + b) + f(a b) + [f(a + b) f(a b)] b /b}

where b = |b| and f some function.