quantum field theory - Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density

$${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$

of the complex scalar field, we have the field

$$\phi(x)=\int{\frac{d^3 \vec{p}}{2(2\pi)^3\omega_\vec{p}}}(a(\vec{p})e^{-ipx}+b^{\dagger}(\vec{p})e^{+ipx}).$$

I'm trying to now find an equation for the $a(\vec{p})$ and $b(\vec{p})$ (with the final goal of finding an expression for $[a(\vec{p}),b^{\dagger}(\vec{q})]$ using $[\phi(x),\Pi(y)]$).

What I should note is that we are considering all of this in the Schrodinger picture (t=0) so I suppose the very first thing to do is change all the $x$'s to $\vec{x}$ right?

The strategy I'm struggling to implement and failing at many points along the way:

1. 
   

   Find the momentum $\Pi^{\phi}(x)=\frac{\partial L}{\partial \dot{\phi}}=\dot{\phi^{\dagger}}$.

   
   


2. 
   

   Add some combination of $\phi(x)$ and $\Pi(x)$ to get rid of one of the creation/annihilation operators.

   
   


3. 
   

   Do an inverse Fourier transform to find $a(\vec{p})$ in terms of $\phi(x)$, for example.

   
   

None of the major text books seem to actually carry this through, and instead write something like "and it's easy to show…". However I don't find it too easy, especially part 3. as I'm not a Fourier transform expert.

Could anyone either direct me somewhere where the above is computed explicitly (in more than 2/3 lines), or help me understand each of the 3 steps above?

(I realise it is easy to find a reference where this is done for the real scalar field, in which case we have $a(\vec{p})$ and $a^{\dagger}(\vec{p})$. Even still, I find it hard to follow parts.)