22.13. ANALYTIC FUNCTIONS 741
=1
2πi
∫C1
f(x+w1h1 +∑
lm=2 zmhm
)(w1− z1)
dw1
=
(1
2πi
)2 ∫C1
1w1− z1
·
∫C1
f(x+w1h1 +w2h2 +∑
lm=3 zmhm
)(w2− z2)
dw2dw1 =(1
2πi
)l ∫C1
· · ·∫
C1
f (x+w1h1 +w2h2 + · · ·+wlhl)
∏lm=1 (wm− zm)
dwl · · ·dw1.
Consider the case when l = 2.(1
2πi
)2 ∫C1
∫C1
f (x+w1h1 +w2h2)
(w1− z1)(w2− z2)dw2dw1 =
(1
2πi
)2 ∫C1
∫C1
f (x+w1h1 +w2h2) ·
∞
∑k2=0
zk22
wk2+12
∞
∑k1=0
zk11
wk1+11
dw2dw1 =
(1
2πi
)2 ∞
∑k2=0
∞
∑k1=0
(∫C1
∫C1
f (x+w1h1 +w2h2)
wk2+12 wk1+1
1
dw2dw1
)zk2
2 zk11 .
Similarly, for arbitrary l, and letting C be any circle centered at 0 with radius smaller thanδ
l ,
f
(x+
l
∑m=1
zmhm
)=
∞
∑kl=0· · ·
∞
∑k1=0
ak1···kl (x,hl , · · · ,h1)zk11 · · ·z
kkl (22.13.35)
whereak1···kl (x,hl , · · · ,h1)
=
(1
2πi
)l ∫C· · ·∫
C
f(x+∑
lm=1 wmhm
)∏
lm=1 wkm+1
mdw1 · · ·dwl . (22.13.36)
Lemma 22.13.2 Let l ≥ 1 and let tm ∈ C. Then if h ∈ X l , then whenever |z| is smallenough, 22.13.35 holds. Also the coefficients satisfy
ak1···kl (x, tlhl , · · · , t1h1) =
(l
∏m=1
tkmm
)ak1···kl (x,hl , · · · ,h1) (22.13.37)
and ∥∥ak1···kl (x,hl , · · · ,h1)∥∥≤C
l
∏m=1∥hm∥ (22.13.38)
for some constant C.