58.1. PERIODIC FUNCTIONS 1841

Lemma 58.1.20 limb→∞ λ (a+ ib)e−iπ(a+ib) = 16 uniformly in a ∈ R.

Proof: From 58.1.30 and Lemma 58.1.19, this lemma will be proved if it is shown

limb→∞

(2

cos2(π( 1

2

)(a+ ib)

) − 2sin2 (

π( 1

2

)(a+ ib)

))e−iπ(a+ib) = 16

uniformly in a ∈ R. Let τ = a+ ib to simplify the notation. Then the above expressionequals  8(

ei π2 τ + e−i π

2 τ

)2 +8(

ei π2 τ − e−i π

2 τ

)2

e−iπτ

=

(8eiπτ

(eiπτ +1)2 +8eiπτ

(eiπτ −1)2

)e−iπτ

=8

(eiπτ +1)2 +8

(eiπτ −1)2

= 161+ e2πiτ

(1− e2πiτ)2 .

Now ∣∣∣∣∣ 1+ e2πiτ

(1− e2πiτ)2 −1

∣∣∣∣∣ =

∣∣∣∣∣ 1+ e2πiτ

(1− e2πiτ)2 −(1− e2πiτ

)2

(1− e2πiτ)2

∣∣∣∣∣≤

∣∣3e2πiτ − e4πiτ∣∣

(1− e−2πb)2 ≤

3e−2πb + e−4πb

(1− e−2πb)2

and this estimate proves the lemma.

Corollary 58.1.21 limb→∞ λ (a+ ib) = 0 uniformly in a ∈ R. Also λ (ib) for b > 0 is realand is between 0 and 1, λ is real on the line, l2 and on the curve, C and limb→0+ λ (1+ ib)=−∞.

Proof: From Lemma 58.1.20,∣∣∣λ (a+ ib)e−iπ(a+ib)−16∣∣∣< 1

for all a provided b is large enough. Therefore, for such b,

|λ (a+ ib)| ≤ 17e−πb.

58.1.28 proves the assertion about λ (−bi) real.By the first part, limb→∞ |λ (ib)|= 0. Now from 58.1.24

limb→0+

λ (ib) = limb→0+

(1−λ

(−1ib

))= lim

b→0+

(1−λ

(ib

))= 1. (58.1.31)