Calculating high-order derivatives of a function like
can be very messy. A useful theorem reduces the calculation
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 +x2/2 +x3/3! ... . The typical term will be . This term is a homogeneous polynomial in the bi 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 .
© Copyright 2001, American Mathematical Society