Let $f(z) = \sin(z),$ viewed as a function in the hardy space $H^1.$ How do I factor $f$ into inner and outer factors? I know that the formula for the outer factor is $$Q_f(z)= \exp(\dfrac{1}{2\pi} \int_{-\pi}^\pi \dfrac{e^{it}+z}{e^{it}-z} \log|\sin(e^{it})| dt)$$ for $z$ in the open unit disk.
This looks like a mess to compute. Is there a way to explicitly find the inner and outer factors of $\sin(z)?$
Obviously, we need to factor out the zeros of $\sin(z)$ in the unit disk, and there is precisely one zero in the unit disk, namely at $0.$So the inner factor $M_f$ contains $z.$ Now, I believe that since $\sin(z)/z$ is analytic in a larger disk than the unit disk, $\sin(z)/z$ is already an outer function. Does this mean that $Q_f(z) = \sin(z)/z?$
How does one in general find the outer and inner factors of a function $f \in H^1?$ What are some good examples to keep in mind?