How does one get the first few terms of the $S$-matrix expansion? According to a set of notes I'm reading
$$\langle p_f | S | p_i \rangle = \delta(p_f-p_i) + 2 \pi \delta(E_f-E_i) \bigg[\langle p_f | V | p_i \rangle + \cdots\bigg]. \tag{1.29}$$
I don't understand where the $2 \pi \delta$ factor comes from in the second term. Could someone please help me see this?
(I'm trying to understand Eq. 1.29 here: http://www.physics.umd.edu/courses/Phys851/Luty/notes/renorm.pdf)
 A: The Hamiltonian is: $H=H_0+V$, where V is the interaction part. The scattering matrix (expansion form to the first order) is:
$$S=1-i\int_{-\infty}^{\infty}V(\tau)d\tau+...$$
where $V(\tau)=e^{iH_0\tau}Ve^{-iH_0\tau}$ (interacting formalism). So:
$$S^{(1)}_{fi}=-i\int_{-\infty}^{\infty}\langle\psi_f|e^{iH_0\tau}Ve^{-iH_0\tau}|\psi_i\rangle d\tau=2\pi\int_{-\infty}^{\infty}e^{i(E_f-E_i)\tau}\langle\psi_f|V|\psi_i\rangle d\tau=2\pi\delta(E_f-E_i)\langle\psi_f|V|\psi_i\rangle$$
(we can do for other terms similarly).
In physical sense, this Dirac delta function guarantees the energy conservation of the system (so in Feynman diagrams, there is always a Dirac delta fucntion in each vertex function ).
A: The first expansion term is $\langle p_f | V |p_i\rangle$. If you assume energy and momentum conservation between initial and final state, namely $E_f = E_i$ and $p_f=p_i$ then the only possibility for the above to be non-zero is in correspondence of 
$$
\delta(E_f-E_i)\delta(p_f-p_i)\langle p_f | V |p_i\rangle = \delta(E_f-E_i)\langle p_f | V |p_f\rangle.
$$
