Page 30 -

To see this, let us return to (2) and conclude that
∞
1 1 |z| j Riemann Sums 23
1
1 1
∞

log Fàz) −
1 − z j 1 −|z| j 1 −|z| j j
j 2 j 2

1 π 2
− 1 ,
1 −|z| 6
or

1 π 2 1
Fàz) exp + − 1 , à 8)
|1 − z| 6 1 −|z|
an estimate which is just what we need. It shows that, away from 1,
where 1 is smaller than 1 , Fàz) is rather small.
|1−z| 1−|z|
Thus, for example, we obtain
1 |1 − z|
Fàz) exp when ≥ 3. à 9)
|1 − z| 1 −|z|
Also, in this same region, setting

∞
2
1 − z π 1 + z
n
φ(z) exp q(n)z , à 10)
2π 12 1 − z
n 0

2 π 2 2 π 2 2
φ(z) exp exp
2π 12 1 − z 12 3(1 −|z|)
so that

1 |1 − z|
φ(z) exp when ≥ 3. à 11)
1 −|z| 1 −|z|
The Cauchy Integral. Armed with these preparations and the
feeling that the coefficients of theelementary functionφ(z)areacces-
sible, we launch our major Cauchy integral attack. So, to commence
the firing, we write

1 Fàz) − φ(z)
p(n) − q(n) dz (12)
2πi C z n+1