I'm interested in constructing a nice proof for my thesis of Wick's Lemma. I'm going to state it in a slightly different context (one which is more useful for my purposes).
Let $Z_b = \displaystyle{\int_{\mathbb{R}^N} e^{-\frac{1}{2}(x^t A x) + b^t x} d^N x} = \sqrt{\frac{(2\pi)^N}{\mbox{det}(A)}}e^{\frac{1}{2}b^tA^{-1}b}$
Then we can state Wick's Lemma as follows:
$\displaystyle{<x_{i_1} ... x_{i_n}>= \frac{\partial}{\partial b_{i_1}}...\frac{\partial}{\partial b_{i_n}}e^{\frac{1}{2}b^t A^{-1}b}|_{b=0}} = \sum A^{-1}_{i_{p_1},i_{p_2}} ... A^{-1}_{i_{p_{n-1}}, i_{p_n}}$
where the sum is taken over all pairings of $(i_{p_1}, i_{p_2}), ... , (i_{p_{n-1}},i_{p_n})$ of $i_1, ... i_n$.
I know this is an induction proof and there is a very nice version of it in "An Introduction to Relativistic Quantum Field Theory" by S. Schweber, but my copy of the book is in the mail. Just wondering if anyone could guide me on this. Thank you.