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II. The Partition Function
26
or
√

∞ 2 2 π 2
b
e −a t e +2abt dt e .
a
−∞
2 π 2 1 2
Thus if we set b and a 1 − z (thinking of z as real
6 1−z
(|z| < 1) for now), we obtain
√
∞ 2 2 2 π π 1
√ 2
zt
e e +π 3 t−t dt √ exp ,
1 − z 6 1 − z
−∞
which gives, finally,
2
e −π /12 ∞ zt 2 π √ 2 t−t 2
φ(z) √ (1 − zð e e 3 dt. (19)
π 2 −∞
Equating coefficients therefore results in

2

e −π /12 ∞ t 2n t 2n−2 π √ 2/3 t−t 2
q(n) √ − e dt (20)
π 2 −∞ n! (n − 1)!
the “formula” for q(n) from which we can obtain asymptotics.
√
Reasoning that the maximum of the integrand occurs near t n
√
we change variables by t s + n, and thereby obtain
2
∞ π

−2 s− √
K n (s)2se 2 6 ds, (21)
q(n) C n
−∞
where
√
e π 2n/3 n n+ 1 2
C n √ ,
n
π 2n e n!
s 2n
1 + √ s 2
−s
2 n √ + s
1 + √ e n 2n .
2
K n (s)

s n
1 + √
n
Since K n (s) → 1, we see, at least formally, that the above integral
approaches
2

∞ π ∞ π
−2 s− √ −u 2
2se 2 6 ds u + √ e du,
2 3
−∞ −∞