Topics In Analysis

52.2.2 The Laurent Series

The Laurent series is like a power series except it allows for negative exponents. First here is a definition of what is meant by the convergence of such a series.

Proof: Let R₁ <

= r₁ −δ < r₁. Then ∑ _n=1^∞a_−n⁻ⁿ converges and so

lim |a−n||w − a|−n = lim |a−n|(r1 − δ)−n = 0 n→∞ n→ ∞

which implies that for all n sufficiently large,

|a−n|(r1 − δ)−n < 1.

Therefore,

∞∑ −n ∞∑ −n n − n |a−n||z − a| = |a−n|(r1 − δ) (r1 − δ) |z − a| . n=1 n=1

Now for

≥ r₁,

|z − a|−n ≤ -1n r1

and so for all sufficiently large n

n |a ||z − a|−n ≤ (r1 −-δ)-. −n rn1

Therefore, by the Weierstrass M test, the series, ∑ _n=1^∞a_−n

⁻ⁿ converges absolutely and uniformly on the set

{z ∈ ℂ : |z − a| ≥ r1}.

Similar reasoning shows the series, ∑ _n=0^∞a_n

ⁿ converges uniformly on the set

{z ∈ ℂ : |z − a| ≤ r }. 2

This proves the Lemma.

Proof: Let R₁ < r₁ < r₂ < R₂ and define γ₁

≡ a + e^it and γ₂ ≡ a + e^it for t ∈ and ε chosen small enough that R₁ < r₁ − ε < r₂ + ε < R₂.

Then using Lemma 52.2.2, if z

ann then

n(− γ1,z)+ n (γ2,z) = 0

and if z ∈ ann

,

n(− γ1,z)+ n (γ2,z) = 1.

Therefore, by Theorem 50.7.19, for z ∈ ann

( ) -1-∫ f-(w) ∞∑ -z −-a n = 2πi γ w − a w − a dw + 2 n=0

∫ ∞ ( )n -1- f-(w)-∑ w-−-a dw. (52.2.7) 2πi γ1 (z − a)n=0 z − a

(52.2.7)

From the formula 52.2.7, it follows that for z ∈ann

, the terms in the first sum are bounded by an expression of the form Cⁿ while those in the second are bounded by one of the form Cⁿ and so by the Weierstrass M test, the convergence is uniform and so the integrals and the sums in the above formula may be interchanged and after renaming the variable of summation, this yields

∞ ( ∫ ) f (z) = ∑ -1- --f-(w-)--dw (z − a)n + n=0 2πi γ2 (w − a)n+1

( ) −∑ 1 -1-∫ ---f (w)-- n 2πi γ (w − a)n+1 (z − a) . (52.2.8) n=−∞ 1

(52.2.8)

Therefore, by Lemma 52.2.3, for any r ∈

,

( ) ∑∞ 1 ∫ f (w ) n f (z) = 2πi γ (w−-a)n+1dw (z − a) + n=0 r

( ) −∑ 1 1 ∫ f (w) n 2πi (w-−-a)n+1 (z − a) . (52.2.9) n=−∞ γr

(52.2.9)

and so

∑∞ ( ∫ ) f (z) = -1- --f-(w-)--dw (z − a)n . n=− ∞ 2πi γr (w− a)n+1

where r ∈

is arbitrary. This proves the existence part of the theorem. It remains to characterize a_n.

If f

= ∑ _n=−∞^∞a_nⁿ on ann let

∑n fn (z) ≡ ak(z − a)k. (52.2.10) k=−n

(52.2.10)

This function is analytic in ann

and so from the above argument,

∑∞ ( ∫ ) fn(z) = -1- --fn-(w-)k+1-dw (z − a)k. (52.2.11) k=−∞ 2πi γr(w − a)

(52.2.11)

Also if k > n or if k < −n,

( ) -1-∫ --fn(w)--- 2πi γr (w − a)k+1dw = 0.

and so

n ( ∫ ) f (z) = ∑ -1- --fn-(w-)--dw (z − a)k n k= −n 2πi γr(w − a)k+1

which implies from 52.2.10 that for each k ∈

,

∫ -1- --fn-(w-)--dw = ak 2πi γr(w − a)k+1

However, from the uniform convergence of the series,

∞∑ n an(w − a) n=0

and

∞ ∑ a (w − a)− n n=1 −n

ensured by Lemma 52.2.5 which allows the interchange of sums and integrals, if k ∈

,

because if l > n or l < −n,

∫ al(w-−-a)l γr(w − a)k+1 dw = 0

for all k ∈

. Therefore,

∫ ak =-1- --f-(w)k+1dw 2πi γr(w − a)

and so this establishes uniqueness. This proves the theorem.