Derivation of Sudarshan-Glauber P function

Homework Statement

It is not a problem, But I am doing a coursework on Quantum Optics on my own. The following derivation is for P-function in Quantum optics from the book Quantum Optics by Marlan O.Scully and M. Suhail Zubairy. I attached the Image of the derivation with this Question.

Homework Equations

Relevant Equations are given in the Image

The Attempt at a Solution

I expanded the trace in the (3.1.11) in |α> basis. But I am unable to get (3.1.13).

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• P-function.jpg
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nrqed
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Homework Statement

It is not a problem, But I am doing a coursework on Quantum Optics on my own. The following derivation is for P-function in Quantum optics from the book Quantum Optics by Marlan O.Scully and M. Suhail Zubairy. I attached the Image of the derivation with this Question.

Homework Equations

Relevant Equations are given in the Image

The Attempt at a Solution

I expanded the trace in the (3.1.11) in |α> basis. But I am unable to get (3.1.13).
They are just saying that (by definition),

$$\int d^2 \alpha ~~\delta(a- \alpha) \delta(a^\dagger - \alpha^*) f(\alpha) f(\alpha^*) = f(a) f(a^\dagger)$$

Yeah I absolutely understand that, but I don't know how to expand f(a)f(a†). Can you help me with this?

nrqed