58.1. PERIODIC FUNCTIONS 1837
From the original formula for ℘,
℘
(1+ τ
2
)−℘
(τ
2
)=
1( 1+τ
2
)2 −1(τ
2
)2 + ∑(k,m)̸=(0,0)
1(k− 1
2 +(m− 1
2
)τ)2 −
1(k+(m− 1
2
)τ)2
= ∑(k,m)∈Z2
1(k− 1
2 +(m− 1
2
)τ)2 −
1(k+(m− 1
2
)τ)2
= ∑(k,m)∈Z2
1(k− 1
2 +(m− 1
2
)τ)2 −
1(k+(m− 1
2
)τ)2
= ∑(k,m)∈Z2
1(k− 1
2 +(−m− 1
2
)τ)2 −
1(k+(−m− 1
2
)τ)2
= ∑(k,m)∈Z2
1( 12 +(m+ 1
2
)τ− k
)2 −1((
m+ 12
)τ− k
)2 . (58.1.21)
Similarly,
℘
(12
)−℘
(τ
2
)=
1( 12
)2 −1(τ
2
)2 + ∑(k,m)̸=(0,0)
1(k− 1
2 +mτ)2 −
1(k+(m− 1
2
)τ)2
= ∑(k,m)∈Z2
1(k− 1
2 +mτ)2 −
1(k+(m− 1
2
)τ)2
= ∑(k,m)∈Z2
1(k− 1
2 −mτ)2 −
1(k+(−m− 1
2
)τ)2
= ∑(k,m)∈Z2
1( 12 +mτ− k
)2 −1((
m+ 12
)τ− k
)2 . (58.1.22)
Now use 58.1.19 to sum these over k. This yields,
℘
(1+ τ
2
)−℘
(τ
2
)= ∑
m
π2
sin2 (π( 1
2 +(m+ 1
2
)τ)) − π2
sin2 (π(m+ 1
2
)τ)
= ∑m
π2
cos2(π(m+ 1
2
)τ) − π2
sin2 (π(m+ 1
2
)τ)
58.1. PERIODIC FUNCTIONS 1837From the original formula for g,(58.1.21)Similarly,1 1 rl rl=a Tat a7 7(5) 5)" &m\F(0.0) (K-53 +m)” (k+ (m—3)7)1 1(mez? (k— 4 +mt)” (k+(m—35) t)”1 1wumecs (R—$—me)? (b+ (m=) 9)i 1~ - (58.1.22)eee (£4+mt—k)° ((m+4 tk)Now use 58.1.19 to sum these over k. This yields,o(5*)-0(5)