Calculating highorder derivatives of a function like
can be very messy. A useful theorem reduces the calculation
to combinatorics.
Wick's theorem
Let us calculate a couple of examples.
To begin, it is useful to write with (the sums running from 1 to d) using the series expansion exp x = 1 + x +x^{2}/2 +x^{3}/3! ... . The typical term will be . This term is a homogeneous polynomial in the b^{i} of degree 2n
Differentiating k times a homogeneous polynomial of degree 2n and evaluating at zero will give zero unless k = 2n. So the job is to analyze the result of 2n differentiations on .
The differentiation carried out most frequently in these calculations is
In what follows will be abbreviated as .
Similarly:

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