## integral form of delta function and Lagrangian multipliers

I have difficulties following the mathematics in the following paper:
http://iopscience.iop.org/0295-5075/81/2/28005
(For those who have no access, please kindly refer to here: https://dl.dropbox.com/u/7383429/Bia...0ensembles.pdf)

To be specific, I have no idea on the derivation from equation (6) to (7):
from
$$Z_1=\sum_{\{a_{ij}\}}\prod_i\delta\left(k_i-\sum_ja_{ij}\right)e^{\sum_{i<j}h_{ij}a_{ij}}$$
to
$$Z_1=\int\mathcal{D}\omega\,e^{-\sum_i\omega_ik_i}\prod_{i<j}(1+e^{\omega_i+\omega_j+h_{ij}})$$
,where $$\mathcal{D}\omega=\prod_id\omega_i/(2\pi)$$

The author said, to derive from (6) to (7), you need to express "the deltas in the integral form with Lagrangain multipliers ωi for every i=1,...,N".

First, as far as I know, to express delta function in the integral form, we will have to introduce the imaginary i to the integral, but there is no imaginary i in the integral. Second, I don't know how did the author apply the Lagrange multiplier since equation (7) does not resemble typical usage of Lagrange multiplier. Third, there is no upper limit and lower limit in the integral, which makes understanding even more difficult.

Is there anyone who can show me the derivation steps?

Thanks!

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