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II. The Partition Function
22
The Approximation. We have prepared the way for the useful ap-
proximation to our generating function. All we need to do is combine
1
(1), (3), and (6), replace w by log , and exponentiate. The result is
z
∞
1
1 − z k
k 1

1 − z π 2
exp [1 + O(1 − zð ]
2π 6 log 1
z
|1 − z|
in ≤ c.
1 −|z|
But we perform one more “neatening” operation. Thus log 1 is
z
an eyesore! It isn’t at all analytic in the unit disc, we must replace
it (before anything good can result). So note that, near 1, log 1
z
(1−zð 2 (1−zð 3 1−z 3 1
(1 − zð + + + ··· 2 + O((1 − zð ),or 1
2 3 1+z log
z
1 1+z + O(1 − zð . Finally then,
2 1−z
∞
1
1 − z k
k 1

2
1 − z π 1 + z
exp [1 + O(1 − zð ] (7)
2π 12 1 − z
|1 − z|
in ≤ c.
1 −|z|

This is our basic approximation. It is good near z 1, which
we have decided is the most important locale. Here we see that
we can replace our generating function by the elementary function

1−z exp π 2 1+z whose coefficients should then prove amenable.
2π 12 1−z
However, (7) is really of no use away from z 1, and, since
Cauchy’s theorem requires values of z all along a closed loop sur-
rounding 0, we see that something else must be supplied. Indeed we
will show that, away from 1, everything is negligible by comparison.