Construction of the phase operator using logarithm of the annihilation operator

We investigate a lema that excludes existence of the phase operator and present a condition to avoid the lema. A method for construction of an analytic function 'f' of the annihilation operator 'a' is given. f(z) is analytic on some compact domain that does not separate the complex plane. Using these results we obtain ln a. Since [N ,-i ln a]=i, we can use ln a to construct an operator $\Phi$, which satisfies the definition of the phase operator.