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1
n+1 dz, an integral rather than a differential operator!
a n
2πi C z fàz) The Approximation 19
Surely this is a more secure approach, because integral operators are
bounded, and differential operators are not. The price we pay is that
of passing to the complex numbers for our z’s. Not a bad price, is it?
So let us get under way, but armed with the knowledge that the
valuable information about fàz) will help in getting a good approx-
fàz) 1
imation to n+1 dz. But a glance at the potentially explosive
C z z n+1
shows us that C had better stay as far away from the origin as it can,
i.e., it must hug the unit circle. Again, a look at our generating func-
n
tion p(n)z shows that it’s biggest when z is positive (since the
coefficients are themselves positive). All in all, we see that we should
seek approximations to our generating function which are good for
|z| near 1 with special importance attached to those z’s which are
near +1.

The Approximation

∞ 1
Starting with (1), Fàz) k , and taking logarithms, we
k 1 1−z
obtain
∞ ∞ ∞ kj
1 z
log Fàz) log
1 − z k j
k 1 k 1 j 1
∞ ∞ ∞ j
1 1 z
z jk . à 2)
j j 1 − z j
j 1 k 1 j 1
Now write z e −w so that w> 0 and obtain log Fàe −w )
∞ 1 1 . Thus noticing that the expansion of 1 begins with

x
k 1 k e kw −1 e −1
1 1 + c 1 x + ··· or equivalently (near 0) 1 e −x + cx + ···,
x − 2 x − 2
we rewrite this as

1 1 e −kw
log Fàe −w ) −
k kw 2

1 1 1 e −kw
+ − + (3)
k e kw − 1 kw 2