We can start with problems that are not fully solved in the text.
Some terms of
as the full expression of . Expanding in powers of and we get:
Now we can get the asked for terms. To get the and term, we need to remember that the left side will apply four derivatives over the power of . For this reason, we need to get the , term:
Let’s do now the and term ( and ):
Now we can try to get the general rules for and term. This requires setting and :
Checking for and :
matching the previously obtained result.
Let’s think this value in terms of an associated graph, with internal vertices of degree 4 and external vertices of degree 1. The number of edges will be , giving this formula for the term:
Expanding in powers of :
where is a constant, independent from and .
The terms cannot be exactly evaluated, but they can be expanded as a series in . Following the book, let’s calculate the term of :
Applying the equation I.2.18, we get: